This is a question about faith, or "seeing" Christ. Faith does not mean that you believe something improbable or unlikely. Faith refers to perceiving something through spiritual sight, through the eyes of the spirit.
manifest in the translation you cited just means "to reveal". It's not a fancy word. This whole passage is a promise that Jesus is making to reveal himself to us so that we will see him, so that we will know that he is living in us.
Here is this promise, in the LEB translation:
“I will not leave you as orphans; I am coming to you. Yet a little time and the world will see me no longer, but you will see me; because I live, you also will live. On that day you will know that I am in my Father, and you are in me, and I am in you. The one who has my commandments and keeps them—that one is the one who loves me. And the one who loves me will be loved by my Father, and I will love him and will reveal myself to him.”
So there are two things happening:
Let's go through this process:
The Process of Seeing
As with all spiritual things, I believe, this process can take many forms, but the scriptural process of faith is that first we hear, and what we hear we see (in spirit), and what we see we believe. It all starts with hearing. "Hear O Israel"
Consequently, faith comes by hearing, and hearing through the word about Christ.
Romans 10.17
It is this strange thing, that God was pleased through the foolishness of preaching to save those who believe. (1 Cor 1.21), but this process happens through preaching and hearing what is preached. And hopefully the subject of the sermon is our place in Christ and his place in us, which is what all sermons should be about.
This is why in Isaiah God promises to plug up the ears of the Israelites so as to prevent them from hearing. But if our ears are not plugged up, if when we hear the scriptures, then something within us testifies "amen" to the word that is spoken, then we are truly hearing, and we are participating in the process of first hearing, then seeing, and then believing our place in Christ and Christ's place within us:
For you have not received a spirit of slavery leading to fear again,
but you have received the Spirit of adoption, by whom we cry out,
“Abba! Father!” The Spirit himself confirms to our spirit that we are
children of God, and if children, also heirs—heirs of God and fellow
heirs with Christ, if indeed we suffer together with him so that we
may also be glorified together with him. Romans 8.15-17
So that is the process. We hear the word of God, the Spirit confirms with our spirit our position in Christ. We hear the word of God, and it happens again. It's not just about hearing words being spoken, but the Spirit has to confirm. I'm sure you've experienced it -- some sermons are just information, but at other times, something rises in you to say "yes!". That is the hearing discussed here. When God said he would stop the ears of the Isreaelites, he was not referring to hearing disabilities, but quenching that affirmation in their hearts.
It is from the testimony of those two witnesses, where the Spirit of God testifies in agreement with our spirit, that faith is born as we "see" with spiritual sight.
But the problem is often we stop there, and so faith is stunted. We skip the "testing" part. But all through scripture God always requires testing to confirm the word that was spoken. Everything must be tested.
The process of Knowing
This is one area of the scripture where Christians are urged to look inward and test themselves:
Test yourselves to see if you are in the faith. Examine yourselves! Or
do you not recognize regarding yourselves that Jesus Christ is in you,
unless you are unqualified? 2 Cor 13.5 (LEB)
The word "test" here, πειράζω, is also interpreted "temptation", which has two meanings, but the appropriate meaning here is:
to endeavor to discover the nature or character of someth. by testing,
try, make trial of, put to the test
Arndt, W., Danker, F. W., Bauer, W., & Gingrich, F. W. (2000). A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed., p. 792). Chicago: University of Chicago Press.
And the word "recognize" is ἐπιγινώσκω which is a synonym for "know". If we test ourselves, we will know that Jesus Christ is in us. But it comes from testing.
Note that we are not testing whether Christ is living in us, but whether we know that he is living in us. It is faith that is being tested. It must be tested. So this is very important - Paul commands the believers to do it. And it has a sense of urgency.
In Psalm 132:4-5, the Psalmist is writing that he will not go to sleep until he finds the dwelling place of Yahweh, but speaking in the spirit he is also talking to us, that we should not stop to rest until we know where YHWH lives, which is within us:
I will give no sleep to my eyes, no slumber to my eyelids, until I
find a place for Yahweh, a dwelling place for the Mighty One of
Jacob.
Now I cannot give you a recipe for how to test yourself, I can only tell you to do it with integrity and sincerity. It was much easier during the time of the early church, as people living together in communion were constantly being tested. Persecution also results in testing. So does poverty or lack in this world. It is harder for a rich man to enter heaven, but with God all things are possible.
But what I can say is that in order to pass from "believe" to "know", that process of testing is required.
Assistance from the Father
For those who are in a situation where they are still uncertain, we have surety in more promises given to us by Christ:
We can ask the father to reveal that Christ is in us
“Ask and it will be given to you; seek and you will find; knock and it
will be opened for you. For everyone who asks receives, and the one
who seeks finds, and to the one who knocks it will be opened. Matt
7.7-8
If we come to the Father and ask, then he will answer. That answer may involve quite a bit of testing. But He will answer. Just as Christ promises to reveal himself to us, and he promises not to turn us away:
John 6.36-40
But I said to you that you have seen me and do not believe. Everyone
whom the Father gives to me will come to me, and the one who comes to
me I will never throw out, because I have come down from heaven not
that I should do my will, but the will of the one who sent me. Now
this is the will of the one who sent me: that everyone whom he has
given me, I would not lose any of them, but raise them up on the last
day. For this is the will of my Father, that everyone who looks at the
Son and believes in him would have eternal life, and I will raise him
up on the last day.”
The only requirements placed on us in the course of this seeking and asking is to do it in integrity and honesty.
God is spirit, and the ones who worship him must worship in spirit and
truth.” John 4.24
Here, "worship" just means "bow down to", or "revere". When we come to the Father to ask him, we bow down to him, and this must be done in spirit and truth.
If we do this, then the Holy Spirit will reveal to us and confirm in our hearts that Jesus Christ is in us - we need only ask the Father and he will accomplish this work within us.