As far as I’m aware, the Truecrypt backdoor thing was speculation regarding the termination of the Truecrypt project, but it was not confirmed. You can see here that development of Truecrypt ceased in 2014. Veracrypt was forked around that time. As for whether or not you can trust it, you’ll want to evaluate the audits that have been performed and decide if you trust them. You can find a link to what seems like the latest audit here.

Truecrypt is abandoned, and has vulnerabilities that will not be fixed. Veracrypt is a fork of Truecrypt that is still actively maintained.