How do Christains argue that "messianic psalms" are messianic and refer to Jesus Christ? I've heard Christians assert that you can prove that Jesus Christ is the Messiah from the book of psalms. There doesn't seem to be any watertight argument for the existence of messianic psalms other than the New Testament says so, let alone that Jesus Christ is the Messiah referred to in the psalms.
The interpretation of the Old Testament in the New is an extremely complex subject and I recommend the book referenced in this summary article about the perspectives involved. However, there is significant support from Jewish sources for reading at least a some, but not all, of the Psalms as Messianic apart from the New Testament and according to Jewish (non-Christian) sources - which to be fair was largely written by Jewish writers as well.

Exhibit A: Psalm 2

Cited several times in Acts, Hebrews, and Revelation, Psalm 2 has been interpreted as Messianic by the Babylonian Talmud, Maimonides, and the Midrash on Psalms just to name a few. That these sources do so after the first century suggests that in spite of the temptation to interpret them apologetically as being non-messianic, it is a genuinely Messianic Psalm.

Exhibit B: Psalm 110

Although less conclusive than Psalm 2, an article entitle "Is Psalm 110 a Messianic Psalm"  says that (footnote 32):


  ...even the first-century Jewish leaders considered the reference to [my Lord] in Psalm 110:1 to be prophetic of Messiah and to be used only of the
  Messiah and not of an earthly Davidic king...


This is especially significant because this is the verse most cited by Jesus himself to prove his Messianic identity.

Exhibit C: Psalm 22

On the other side, a previous discussion (here) seems to suggest conclusively that Psalm 22 has not been interpreted Messianic-ally by Jewish interpreters. 

These are just three examples that show there is diversity in the way 'Christian' Messianic Psalms were viewed outside of Christianity. But two final points are worth noting:

(1) As "Dan" in the discussion on Psalm 22 notes, it is often difficult to find actual pre-Christian Rabbinic commentary which means that (a) there is apologetic incentive in later sources not disagree with NT interpretations and (b) significant potential for disagreement about what first century Jews really thought. See this article for the dates of Rabbinic literature.

(2) Considering Jesus, Paul, Stephen, and most of the early church were Jewish - some of them highly educated in the Law - so while we can assume bias on their part, this should not cause us throw out their testimony whole-sale, or discount its Jewish-ness. In fact Acts 17:11 tells us it is reading the scriptures eagerly that convinced the Bereans and many others of the validity of Jesus' claims to be the Messiah. This was before the writing of the NT and therefore means it was through reading directly from the Jewish scriptures that they were convinced.