The bronze mountains represent the entrance or gateway to the presence of God and in particular are reminsicent of the two great bronze pillars of Solomon's temple.
Context of the Canon
Mountains are used often as symbols throughout the Hebrew Bible. Their use is not uniform, but there are identifiable symbolic themes. Mountains may represent kingdoms and their rulers. We see this in a place like Psalm 30:6-7, where David's kingdom is likened to a mountain that he feels cannot be shaken. Similarly in the vision in Daniel 2, the unhewn stone that becomes a mountain to fill the whole earth is a kingdom to crush all other kingdoms.
For our purposes though, the major symbolic link concerning mountains is to the dwelling place of God. This connection can really be elaborated at length. For example, there are notable links between the pattern of the tabernacle and encounter between God and Moses/Israel on Sinai - the court corresponding to the perimeter around the mountain where the people stood, the holy place that only the priests could enter corresponding to the base of the mountain where the 70 elders could come and eat the covenant meal with God, and the holy of holies where God's glory cloud dwelled and that only the high priest could enter corresponding ot the peak of the mountain where the cloud was and where only Moses could approach. But we also see this repeated again and again in the Psalms (e.g. 3:4 15:1, 24:3, etc...) and in the prophets (cf. Isa. 2:2-3, Ezek 40:2f). Mountains were readily seen as the connecting place between earth and the heaven where God dwelled.
Lastly, though, we note that mountains were also understood as the foundations or pillars of the earth (see Job 9:5-6). They are generally imovable but that God might shake them.
Bronze was certainly a symbol of strength. In the Daniel 2 vision, the legs of the statue were of bronze signifying their strength. Similarly in Ezekiel's vision in 1:7 he sees legs of bronze. Isaiah compares the stuborness of the people in 48:4 to having a forehead made of bronze.
Again, though, the major thematic use of bronze in the Hebrew Bible for our purposes is undoubtedly in connection with the dwelling place of God. A great deal of bronze was used in the construction of the tabernacle and later Solomon's temple. 1 Kings 7:47 says that so much bronze was used it couldn't even be weighed. Of particular note is the construction of two massive bronze pillars at the front of the temple.
Context of Zechariah
The vision starting in 6:1 of Zechariah concludes a series of eight visions beginning in chapter 1. 1:1 begins with the word of the Lord coming to the prophet, there is a series of eight visions, and then the word of the Lord again comes to the prophet at the end of chapter six. The vision here in 6:1 can be seen as creating an inclusio with the first vision, and it should be seen as a development of that vision. The purpose of Zechariah's prophecy is to encourage the completion of the second temple by the returned exiles (cf. Ezra 5:1).
The two major themes of the eight visions are 1) the re-establishment of the temple and the priesthood in connection with it through the two annointed for the task (Joshua and Zerubbabel - cf. Ezra 5:2) and 2) judgment on the enemies of Israel for going to far in Israel's punishment. In the opening vision, the prophet sees four colored horses who go throughout the earth and find it at rest when it should not be. Next there is a vision of four horns to be smashed by four craftsmen. And then in chapter two the vision of Jerusalem being measured combines the two themes with the ingathering of the people and the rebuilding of the temple/city alongside with a pronouncement of judgment on the nations (2:9).
Context of the Immediate Pericope
All this leads up to our vision which recalls the four horse riders (scouts), now seen as four chariots (warriors). Here the two themes are linked. Verse 5 explicitly states that the four chariots are the four winds which were standing in the presence of God and are now going out into the world. The vision then ties the two themes of temple and judgment together by showing that the establishment of the new temple will be the means by which God will judge the nations.
The two bronze mountains remind the people of the bronze pillars of Solomon's temple, emphasizing also the strength and dominion of God's house, and therefore encouraging them to rebuild the temple knowing that through it God will answer their pleas: "Lord Almighty, how long will you withhold mercy from Jerusalem and from the towns of Judah, which you have been angry with these seventy years?"