Why does the app I downloaded say that 1 in binary is 00110001?
The app you have converts strings in the ASCII character set to their binary representation.  This representation assigns eight bits for each character in the string. For example, the word "cat" is composed of characters number 99, 97 and 116, which is 01100011 01100001 01110100.

What your app is doing is converting the string "1" into its binary ASCII representation. "1" is character number 49 in ASCII, which is 00110001, and it's pure coincidence that this ends in a&nbsp;1.

You're completely correct that, as a number, 1 in decimal is 1 in binary.