For standard tunes like this, the "changes", or chords are pretty public knowledge -- you just have to know where to find them. Frank is singing in the key of E in this recording, which may or may not be the same as the chords you find, so if you wanted to play along with the recording, you might have to do some transposition.
Turns out you can find a lot of lead sheets right on Google -- the first few results of a search for here's that rainy day lead sheet are scans from a fake book (which are huge repositories of exactly this information, for hundreds of "standard" tunes).
However, that book has the song in the key of G. This means in order to play along with Frank's recording, you will need to transpose each chord down a minor 3rd.
Another tool I personally find very useful is a piece of software called iReal b. Among other things it can be used to store chord changes (which are public domain, by the way) and display them in any transposition. So, I was able to easily pull up the changes for "Here's That Rainy Day" in the key of E:
Do not be concerned if the chords here are slightly different from those in the fake book linked earlier -- it is common to see substitutions or alterations; rarely will either version be outright incorrect.
(Also, I checked these out with the recording -- they should fit together with the strings just fine.)