I'm studying this video, and the speaker starting around 0:38 uses 我都 a number of times (我都覺得，我都說。。。）
I don't really understand this use of 都. Can someone clarify what is going on here and what this means?
The speaker you're referring to has these subtitles following his speech:
所以说真的, 我们每次学的时候, 难怪我都觉得说, 人家看我怪怪的。 我都说(something in Japanese), 可是那个(something in Japanese)是女生的, 对不对？
What I'm actually hearing (I've double checked with speed 0.25) is this:
所以说真的, 我每次学的时候, 难怪我都觉得人家看我的时候怪怪。 我都说(something in Japanese), 可是那个(something in Japanese)是女生的, 对不对？
He speaks at a fast pace, which results in being not completely 标准 (standard), similar to English natives saying 'wanna' instead of want to.
What I hear might also be different from what he actually says, please correct me if I'm wrong.
Answer to your question:
The 都 in '我都说' (which is your question), means 'always'. The speaker doesn't say 每次, but that is exactly what he means. See here for expressing 'always' with 每次...都.
In addition to this, I would like to add to the answer of songyuanyao (I don't have enough reputation to comment).
都 means already, only in the structure: 都...了. See here for further explanation.
I wouldn't try to translate all of the「都」exactly.
This talk show is in the format of an extremely fast, colloquial dialogue, and makes heavy use of filler words/syllables which you wouldn't find in clearly enunciated speech. I don't believe that the first「都」has an exact translation; my attempt would be something like
No wonder I'm like,...
"they're all giving me weird looks"...
whenever I say わたしは...
The first line especially has unnecessary syllables. All that's need to be said is
No wonder I feel like
Under ordinary circumstances,「都」can be translated in the clearly enunciated structures
but this is not the case at the timestamp given in this talk show.