Silseth, 26, is having a career year out of the Angels bullpen, tossing a 2.72 earned run average with 48 strikeouts through 39.2 innings in 47 games.
The Rangers are optioning O'Hoppe, 26, to Triple-A as he is currently hitting a career-worst .209 average and .565 OPS with four home runs and 23 RBI through 75 games.
Both players are set to be free agents after this season.
Los Angeles are receiving 19-year-old shortstop Angel Arredondo, who is slashing .243/.371/.392 with eight home runs and 44 RBI with three steals through 85 games in Single-A.
The Rangers currently sit atop the American League West division with a 55-53 record, sitting just 1.5 games ahead of the Houston Astros (54-55) and two games ahead of the Seattle Mariners (53-55).
Texas gets some bullpen help they desperately need, as they rank 19th in the league with a combined 4.24 ERA. Silseth's 2.72 ERA would be the third-best in the bullpen (min. 30 games) behind Tyler Alexander (2.70) and Jacob Latz (1.80).
As for the Angels (42-66), they continue to be the worst team in the AL and the second-worst team in the league (Colorado Rockies - 42-67).