The definition from Wikipedia about multi-valued treatment is quite ambiguous to me, is there any other simpler definition or intuitive example for this term?
In a standard treatment effects set up there are only two options to compare: Either the subject is treated or the subject is non-treated.
Multi-valued treatment happens when subjects are not only treated but potentially treated in different ways. If there are two pills people can eat they can either eat one or the other or none (assuming eating both is disallowed by construction). In that case there are two treatments and one non-treatment.
Multi-valued treatment can also be ordinal. Assuming one pill is a higher dose of the same than the other pill, then individuals taking the high dose pill are treated 'more' than those who take low dose pill.
Finally, treatment can be continuous when the medicine in the pills are given in continuous sized dose here there are uncountably many potential treatment values.