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.
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.