The definition of pseudofunctors on nLab is actually quite explicit, but it's for pseudofunctors between bicategories. Borceux (Handbook of Categorical Algebra, volume 1, 7.5) gives the definition for strict categories.
As for what a coherent (iso)morphism is, it's a morphism that satisfies coherence laws. These laws ensure that things that ought to be equal (and trivially would be, if the morphisms were identities) really are.
For example in this case you are given isomorphisms $φ : Ff ∘ Fg ≅ F(f ∘ g)$, but to do anything, you'll obviously need isomorphisms $Ff ∘ Fg ∘ Fh ≅ F(f ∘ g ∘ h)$ too, and similarly for any number morphisms.
Of course you can get from $Ff ∘ Fg ∘ Fh$ to $F(f ∘ g ∘ h)$ using $φ$, but you can do so in two ways: via $Ff ∘ F(g ∘ h)$ or via $F(f ∘ g) ∘ Fh$.
One of the coherence laws for pseudofunctors says that these two ways are the same. That given this, all paths (constructed from $φ$) from $F(f_1) ∘ ... ∘ F(f_n)$ to $F(f_1 ... f_n)$ are the same is then a part of the coherence theorem for pseudofunctors (but these can also be stated in more sophisticated ways).
As a side-note, monoidal categories are a good way to get familiar with coherence and coherence theorems. In a way, anybody who uses product of sets or tensor products already is.