Why are some vowel sound not pronounced in Japanese, like the "u" in "su" and the "i" in "shi"?

The rule, in general:

"u" & "i" are not pronounced or barely whispered between unvoiced consonants (ones that don't vibrate the vocal cords. In Japanese, that's: k, s, sh, t, ch, ts, h, p) or when they are the last syllable.

That fits desu and itakimasu if they are the last word, for example. Note though, for example, that in "desu ne", the U is pronounced.

That fits "mashita" as well: "i" between the unvoiced SH and T.

One additional example: hito. The "i" between unvoiced "h"and T. The "h", though, is not a true H sound there. It is like the "h" of English "human", without the UMAN. In English, those two sounds are treated as if they are variants of the same sound (they are not, but the "human" h sound only occurs in the HU combination). Japanese does similar, It's treated as a variant of H, usually HI + another unvoiced consonant.

(in some languages, that sound is a distinct sound from H, and not treated as a variant of it).

It is a common way of speaking. Not truly informal.