い and う sounds often get dropped in speech. I doubt he mentions 私たち specifically, but I’m fairly sure he at least explains the concept, if only to explain why です is “des” and not “desooo”. Also, でした is “deshta”.

I’ll give you a rule of thumb, but don’t worry about it too much. Listen a lot, that will be a better guide then trying to remember any hard and fast rules.

The sounds “i” and “u” are often dropped at the end of words or when they occur between two unvoiced consonant sounds. Since hiragana, can’t be broken down into their constituent sounds, I’ll use romaji to explain.

ashita – a shi ta: the “i” part of “shi” occurs between a “sh” and a “t” both of which are unvoiced. In this case the “I” sound is silent. — ash*ta –
asakusa – a sa ku sa: the “u” part of “ku” is silent because “k” and “s” are both unvoiced sounds. — asak*sa –
tatsu – ta tsu – the “u” part of “tsu” is often silent since it follows and unvoiced “ts” sound at the end of a word. — tats* –
hachi – ha chi – the “i” part of “chi” is silent since it follows an unvoiced “ch” sound at the end of a word. — hach* –

Keep in mind that these are not rules in Japanese. They are merely patterns in speech, similar to how “ed” is pronounced differently in English in the words “passed”, “calmed”, and “heated”.
You would not be wrong to pronounce 私たち with the “i” sounds present. In fact, in song lyrics you can often hear the vowels that might be omitted in speech as the rhythm of the song might cause that syllable to receive an atypical amount of stress.