Public Freakout

I don't know if there's video footage of Padilla specifically preceding this moment of the press conference, but I think this clip captures most of what happened. Here's a description from AP News:

Padilla interrupted the news conference after Noem delivered a particularly pointed line, saying federal authorities were not going away but planned to stay and increase operations to “liberate” the city from its "socialist" leadership.

- https://apnews.com/article/alex-padilla-noem-immigration-protest-california-f67d220a0254473c53c16aa96f554239

In terms of why he was removed, the article also says this:

In a statement, DHS said that Padilla “chose disrespectful political theater” and that Secret Service “thought he was an attacker.” The statement claimed erroneously that Padilla did not identify himself — he did, as he was being pushed from the room.

Which is obviously a retroactive justification. From what I can tell they did it because it was "disrespectful" to Noem (which is not illegal) and they knew they could get away with it.

EDIT: Found some slightly longer footage here which just shows him pressing towards the front of the group before getting pushed back.