Its quite truthworthy. Its run by Troy Hunt (known security researcher) and : "When you search Pwned Passwords
The Pwned Passwords feature searches previous data breaches for the presence of a user-provided password. The password is hashed client-side with the SHA-1 algorithm then only the first 5 characters of the hash are sent to HIBP per the Cloudflare k-anonymity implementation. HIBP never receives the original password nor enough information to discover what the original password was." from https://haveibeenpwned.com/Privacy
My only concern with the site is some privacy implications. I entered a friend's email just to check for him and it wasn't validated at all, and I found out a few sites he had accounts with. Nothing too concerning was revealed, but privacy for its own sake is a valid goal IMO.
As far as I know hibp specifically hides sensitive breaches (such as the Ashley Madison one) to non-verified access. Also, he basically only shows public data; your privacy was already gone back when the original company failed to secure their servers.
Understood, it's a small complaint, the data is already out there on the web and it's not his fault. But there is value in aggregation or the site wouldn't exist. It makes it easier to just put a few emails in there and see what shows up for fun or malice.
It's great that sensitive breaches are apparently hidden but I'd be wary of judging for other people what is sensitive. Some like Ashley Madison are obvious, others less so.
If you have JavaScript enabled, the cleartext password is hashed in the browser and the hash is truncated, and a list based on the truncated hash is retrieved to be checked against - the only information leaked is that you searched for one password amongst many. Read Troy's articles about how fishing is protected against - I have written the above from memory.
You can download all the hash files if you wish to run purely locally.
Also the site hosting Troy's list is Cloudflare. Cloudflare act as a https proxy for a large number of sites, so they already have access to a large number of passwords.
Which shows that it is fairly strongly "unique", since no-one else has used it and been pwned (or he hasn't reused it and been pwned).