She was probably their chaperone

It seems most likely that Madame Maxime simply was accompanying the students to the Quidditch World cup.

At least one other school attended the event, though it might have been more of a think tank or similar organization than a school for young witches and wizards:

Three African wizards sat in serious conversation, all of them wearing long white robes and roasting what looked like a rabbit on a bright purple fire, while a group of middle-aged American witches sat gossiping happily beneath a spangled banner stretched between their tents that read: the salem witches’ institute.

Furthermore, school trips to the FIFA World Cup, on which the Quidditch World Cup is based, are not unheard of:

While most college students were home this summer enjoying some time off from school, a group of 17 students participated in the ultimate field trip for their sport management class — a trip to South Africa that included watching the 2010 FIFA World Cup.

Undergraduate and graduate students from Mason, as well as Indiana University, Miami University of Ohio, Ohio University, Seattle University and Virginia Commonwealth University, spent close to three weeks traveling to different parts of South Africa to study, sightsee and take in World Cup soccer matches.

For wizards, with their much faster means of transportation, going on a school trip to the World Cup would be much more feasible.

She was probably their chaperone

It seems most likely that Madame Maxime simply was accompanying the students to the Quidditch World cup.

At least one other school attended the event, though it might have been more of a think tank or similar organization than a school for young witches and wizards:

Three African wizards sat in serious conversation, all of them wearing long white robes and roasting what looked like a rabbit on a bright purple fire, while a group of middle-aged American witches sat gossiping happily beneath a spangled banner stretched between their tents that read: the salem witches’ institute.

Furthermore, school trips to the FIFA World Cup, on which the Quidditch World Cup is based, are not unheard of:

While most college students were home this summer enjoying some time off from school, a group of 17 students participated in the ultimate field trip for their sport management class — a trip to South Africa that included watching the 2010 FIFA World Cup.

Undergraduate and graduate students from Mason, as well as Indiana University, Miami University of Ohio, Ohio University, Seattle University and Virginia Commonwealth University, spent close to three weeks traveling to different parts of South Africa to study, sightsee and take in World Cup soccer matches.

For wizards, with their much faster means of transportation, going on a school trip to the World Cup would be much more feasible.

In addition, as @BolteAltamont mentions, the Beauxbatons students say

"Ou est Madame Maxime? Nous l'avons perdue -"

Or

"Where is Madame Maxime? We have lost her–"

This seems to indicate that they had been with Madame Maxime previously, adding weight to the idea that she had accompanied them, rather than merely having attended the match. In addition, the students seemed to think it was very important to find her, which would be unlikely had they been there on their own trip.

She was probably their chaperone

It seems most likely that Madame Maxime simply was accompanying the students to the Quidditch World cup.

At least one other school attended the event, though it might have been more of a think tank or similar organization than a school for young witches and wizards:

Three African wizards sat in serious conversation, all of them wearing long white robes and roasting what looked like a rabbit on a bright purple fire, while a group of middle-aged American witches sat gossiping happily beneath a spangled banner stretched between their tents that read: the salem witches’ institute.

Furthermore, school trips to the FIFA World Cup, on which the Quidditch World Cup is based, are not unheard of:

While most college students were home this summer enjoying some time off from school, a group of 17 students participated in the ultimate field trip for their sport management class — a trip to South Africa that included watching the 2010 FIFA World Cup.

Undergraduate and graduate students from Mason, as well as Indiana University, Miami University of Ohio, Ohio University, Seattle University and Virginia Commonwealth University, spent close to three weeks traveling to different parts of South Africa to study, sightsee and take in World Cup soccer matches.

She was probably their chaperone

It seems most likely that Madame Maxime simply was accompanying the students to the Quidditch World cup.

At least one other school attended the event, though it might have been more of a think tank or similar organization than a school for young witches and wizards:

Three African wizards sat in serious conversation, all of them wearing long white robes and roasting what looked like a rabbit on a bright purple fire, while a group of middle-aged American witches sat gossiping happily beneath a spangled banner stretched between their tents that read: the salem witches’ institute.

Furthermore, school trips to the FIFA World Cup, on which the Quidditch World Cup is based, are not unheard of:

While most college students were home this summer enjoying some time off from school, a group of 17 students participated in the ultimate field trip for their sport management class — a trip to South Africa that included watching the 2010 FIFA World Cup.

Undergraduate and graduate students from Mason, as well as Indiana University, Miami University of Ohio, Ohio University, Seattle University and Virginia Commonwealth University, spent close to three weeks traveling to different parts of South Africa to study, sightsee and take in World Cup soccer matches.

For wizards, with their much faster means of transportation, going on a school trip to the World Cup would be much more feasible.