Text messages are going to fall under the hearsay rules, so to use them in court you need to find a hearsay exception or exclusion that applies to allow their introduction. For example, it may be possible to establish that they are statements against interest, or admissions by a party opponent.

The fact that they're in writing makes them potentially more credible than other statements, recalled but not in writing. If you want to try to establish the validity of the text message exchange, you can try to get records from the cell phone service(s) -- noting that you may also need somebody who can testify as to the authenticity of those records -- or get the other party to admit that the transcript of the exchange is accurate. But as with any hearsay statement, once you establish an exception or exclusion, you can get it before the court and it falls on the other party to dispute having made the statement.

For the hearsay rules, generally, see the Federal Rules of Evidence, Article VIII. You need to look at the rules for the state in which the litigation is occurring, as they are the ones that govern litigation in that state, but most states have hearsay rules that very closely resemble the federal rules.