A medium character can probably ride a mule awkwardly, but only a small or tiny character can effectively use a mule as a mount in combat.
The Beast of Burden ability explicitly states exactly what game mechanic it affects - how much weight the mule can carry. It doesn't have any effects beyond that. The mule is strong enough to carry most medium-sized creatures, and that's probably good enough for transportation purposes, but it's just too small for them to be able to effectively ride it as a mount in combat.
Your confusion will hopefully be resolved by looking back at the Ability Scores section of the basic rules, which is where the rules for determining carrying capacity are described:
Carrying Capacity. Your carrying capacity is your Strength score multiplied by 15. This is the weight (in pounds) that you can carry, which is high enough that most characters don't usually have to worry about it.
Size and Strength. Larger creatures can bear more weight, whereas Tiny creatures can carry less. For each size category above Medium, double the creature's carrying capacity and the amount it can push, drag, or lift. For a Tiny creature, halve these weights.
The mule, as medium creature with a strength of 14, would normally have a carrying capacity of 210lbs - but because its special ability means it is treated as large for the purpose of determining carrying capacity, that value is doubled to 420lbs (the value given for the mule in the mounts table).
Not all mules are equal and not all mules are Mules
It may be worth keeping in mind that horses/donkeys/mules are creatures with considerable variation of size between individuals/breed - wikipedia suggests that weight can range between a mere 50lbs to over 1000lbs. One creature stat block will never accurately describe all of them (you may note there are several different varieties of horse to choose from in the creature lists, including the much smaller pony). Whilst we can easily imagine a mule big enough to serve as a combat mount for a human, and such mules undoubtedly exist in the real world - that's not the mule as described by the Mule creature stat block, which presumes a relatively small equine. A larger mule should probably be represented using the statistics for an equivalent horse - the fact that it is actually a mule is not important for the mechanics of the game.