Translation:That is everyone's problem, not just yours!
(More experienced Esperanto-users, feel free to correct me if I am mistaken.) "Via" by itself would be the possessive adjective "your", which would require a noun to describe. When we add the "la" to make "la via", that turns it into a possessive pronoun, and it is able to stand by itself in the sentence.
On another note, I believe in some cases "via" by itself can mean "yours". I imagine this is only in sentence constructions like "La problemo estas via", where it would act as an adjective in conjunction with the verb "esti (to be)"
It does indeed look strange, I would read this as if whoever was saying this phrase was talking about a person.
That (one) is everyone's problem.