The Greek text of Galatians 6:10 reads:
Ἄρα οὖν ὡς καιρὸν ἔχομεν, ἐργαζώμεθα1 τὸ ἀγαθὸν πρὸς
πάντας, μάλιστα δὲ πρὸς τοὺς οἰκείους τῆς πίστεως.2
A literal translation of this passage is:
Consequently therefore, as we have opportunity, let us do good to all,
and especially to those of the household of faith.
The double connective at the beginning of the sentence is emphasizing the conclusion of a line of argument. The author had just warned his readers to help one another when caught in sin and to 'sow to the Spirit' rather than to his flesh so that he might reap eternal life (v. 8). The reader is then encouraged not to give up and become discouraged/weary in doing good. V. 10 emphasizes the conclusion of this line of thought, encouraging the reader to do good to all, but especially to members of the household of faith.
The primary lexeme of interest to the question is μάλιστα, a superlative adverb (adverb of degree) meaning "to an unusual degree, most of all, above all, especially, particularly."3 This passage certainly called its readers to do good to all, but it also singled out "those of the household of faith." This is the plain sense of the Greek and comes out well in the English translation cited and quoted in the question.
This makes sense in light of the context, which called the 'spiritual' readers (v. 1) to 'carry one another's burdens' (v. 2), among other things. Many of these exhortations only make sense within the early Christian community to which Paul was writing, so it is not surprising that he would single them out in his concluding statement. It only seems strange when the verse is taken in isolation, divorced from its context (why would the author exhort his readers to do more good for those in their household of faith than for those outside of it?). But in context the author has specifically defined what he means by 'doing good,' and much of it is particularly applicable only to other 'spiritual' persons (who are members of the household of faith, i.e. early Christians in Galatia).
1 The hortatory subjunctive (ἐργαφώμεθα, 'let us') is the most well-supported reading and has significant external support. "Although the indicative form of the verb ἐργαφόμεθα (we work/we are working) also has the support of several otherwise strong witnesses, the context favors the hortatory subjunctive. In later Greek, the vowels ω and ο were scarcely distinguished in pronunciation."
Roger L. Omanson and Bruce Manning Metzger, A Textual Guide to the Greek New Testament: An Adaptation of Bruce M. Metzger’s Textual Commentary for the Needs of Translators (Stuttgart: Deutsche Bibelgesellschaft, 2006), 383.
2 NA27 — Eberhard Nestle et al., The Greek New Testament, 27th ed. (Stuttgart: Deutsche Bibelgesellschaft, 1993), 502, emphasis mine.
3 μάλιστα is the superlative form of the adverb μάλα. William Arndt, Frederick W. Danker, and Walter Bauer, A Greek-English Lexicon of the New Testament and Other Early Christian Literature (Chicago: University of Chicago Press, 2000), 613.