I’ve heard casters use the term several times in professional games, but I’m not sure precisely what it means.
Similar to other sports, it means the team in the lead needs to keep up the pressure and not let the team defending catch up.
"Keep the tempo up" in regards to LoL generally means engage first, go for global objectives and overall not play passively.
This is because with an early gold lead, your team is generally stronger, so if you keep the tempo going, it's going to generate even more of a lead.
Tempo can also mean a psychological mindset. A team that is behind can "up the tempo" of play, especially when things are going right. A team that is behind can make a huge play and gain tempo off of that. The opposing team can start to think that the other team is better than they are mechanically and globally on the map.
When a team is behind and they make a great play (not just in LoL but other sports) you'll hear there has been a tempo change. The ability to catch up after being behind will often lead to that team winning because they had all the tempo in the remainder of the match.