What is the purpose of 赴 ("to visit") within 全力以赴 ("to make an all-out effort")?
This involves inversion (倒裝) in classical Chinese. The most sensible arrangement in modern Chinese is

以全力赴（之）
To go towards / approach (sth.) with all one's might,

which obviously suggests the meaning 'trying one's best in doing something'. The grammatical object is the target but is omitted (perhaps understood) in the idiom.

While 赴 is transitive, 全力以赴 has more or less become intransitive (see the following example) because it is idiomatic to consider it as a whole that describes a state of being, rather than an approach towards any specific problem (the object) when we parse the idiom.

我會全力以赴的。 I will try my best.