Why are weak lewis bases strong nucleophiles?
Electrophiles are Lewis bases because the two definitions have the same definition in terms of electrons.
In the Lewis definitions of acids and bases, a Lewis acid is defined as an electron pair 'acceptor', which will acquire an electron pair. A Lewis base is anything that gives this electron pair, hence the term 'donor'.
A nucleophile is a chemical species that donates an electron pair to an electrophile to form a chemical bond in relation to a reaction. (https://en.wikipedia.org/wiki/Nucleophile) In other words, a nucleophile is an 'electron loving' chemical.
You can see that the two definitions overlap in terms of giving electrons. A nucleophile will give electron pairs, and thus by definition are Lewis bases, and vise versa.