Trust is a very problematic issue, and that's the essence of why security is difficult in every field, including computers and Internet communication. Do you trust Tails and its developers? Do you think we have planted backdoors in Tails so we can take control of your computer, or that we make Tails generate compromised encryption keys in order to enable the government to spy on you? Do you simply trust our word on that we are legit?
No matter what your opinion is in this matter you should ask how you reached that conclusion. Both trust and distrust need to be established based on facts, not gut feeling, paranoid suspicion, unfounded hearsay or our word. Of course, we claim to be honest, but written assurances are worthless. In order to make an informed decision you must look at the greater picture of what Tails is comprised of, our affiliations, and possibly how others trust us.
Free software, like Tails, enables its users to check exactly what the software distribution consists of and how it functions since the source code must be made available to all who receive it. Hence a thorough audit of the code can reveal if any malicious code, like a backdoor, is present. Furthermore, with the source code it is possible to build the software, and then compare the result against any version that is already built and being distributed, like the Tails ISO images you can download from us. That way it can be determined whether the distributed version actually was built with the source code, or if any malicious changes have been made.
Of course, most people do not have the knowledge, skills or time required to do this, but due to public scrutiny anyone can have a certain degree of implicit trust in Free software, at least if it is popular enough that other developers look into the source code and do what was described in the previous paragraph. After all, there is a strong tradition within the Free software community to publicly report serious issues that are found within software.
The vast majority of all software shipped in Tails comes from the Debian GNU/Linux distribution. Debian is arguably the Linux distribution whose software packages are under the deepest public scrutiny. Not only is Debian itself one of the largest Linux distros, but it's also one of the most popular distros to make derivatives from. Ubuntu Linux, for instance, is a Debian derivative, and the same goes transitively for all of its derivatives, like Linux Mint. Thus there are countless people using Debian's software packages, and countless developers inspect their integrity. Very serious security issues have been discovered (like the infamous Debian SSH PRNG vulnerability), but backdoors or other types of intentionally placed security holes have never been found to our knowledge.
Tails anonymity is based on Tor, which is developed by The Tor Project. The development of Tor is under a lot of public scrutiny both academically (research on attacks and defenses on onion routing) and engineering-wise (Tor's code has gone through several external audits, and many independent developers have read through the sources for other reasons). Again, security issues have been reported, but nothing malicious like a backdoor -- we would argue that it's only uninformed conspiracy theorists that speculate about deliberate backdoors in Tor these days. Furthermore, Tor's distributed trust model makes it hard for a single entity to capture an individual's traffic and effectively identify them.
One could say that Tails is the union of Debian and Tor. What we do, essentially, is gluing it all together. Hence, if you trust Debian and The Tor Project, what remains to establish trust for Tails is to trust our "glue". As has been mentioned, Tails is Free software, so its source code is completely open for inspection, and it's mainly comprised by a specification of which Debian software packages to install, and how they should be configured. While Tails surely doesn't get the same amount of attention as Debian or Tor, we do have some eyes on us from especially the Tor community, and also some of the general security community (see our audits page). Given that Tails' source code is comparably small and devoid of complexities, we're in a pretty good spot compared to many other projects of similar nature. Our specification and design document is a good starting point to understand how Tails works, by the way.
With all this in light (which you ideally also should try to verify), you should be able to make an informed decision on whether or not you should trust our software.