According to the Chronograph of 354, the Colosseum provided seating for around 87,000 people, but that could be kind of high. Most of the modern sources I've seen estimate the number to have been somewhere between 50,000 and 80,000, more or less. Not that it really matters. The city of Rome at its height had a population ranging from one million to as much as four or five million, depending on what sources you read. I don't know how many Romans regularly attended the games, but you can imagine the turmoil that would occur if a hundred-thousand people tried to get in and find a seat.
The architects who designed the amphitheater tried to plan for large crowds, however:
"The Colosseum's huge crowd capacity made it essential that the venue could be filled or evacuated quickly. Its architects adopted solutions very similar to those used in modern stadiums to deal with the same problem. The amphitheatre was ringed by eighty entrances at ground level, 76 of which were used by ordinary spectators. Each entrance and exit was numbered, as was each staircase ... Each tier was divided into sections (maeniana) by curved passages and low walls (praecinctiones or baltei), and were subdivided into cunei, or wedges, by the steps and aisles from the vomitoria [see note]. Each row (gradus) of seats was numbered, permitting each individual seat to be exactly designated by its gradus, cuneus, and number." (Wikipedia)
"Spectators were given tickets in the form of numbered pottery shards, which directed them to the appropriate section and row. They accessed their seats via vomitoria (singular vomitorium), passageways that opened into a tier of seats from below or behind. These quickly dispersed people into their seats and, upon conclusion of the event or in an emergency evacuation, could permit their exit within only a few minutes. The name vomitoria derived from the Latin word for a rapid discharge, from which English derives the word vomit."
This system must have worked pretty well most of the time. I've read about riots breaking out in the Colosseum over political conflicts and the like, but I've never read about the plebs staging a riot because they couldn't get seats. It would be interesting to know if people were routinely barred from entering because the arena was full. The scenes outside the Colosseum might have gotten pretty chaotic in situations like that.
The next video (2018) shows what is left of the gladiator's entrance to the Colosseum. I'm sure this passage was less crowded than the vomitoria the spectators used, but this wasn't a good way to get into the arena.