"Why is this form correct?" I'm not sure that it *is* technically correct, but it sounds natural in the US, particularly in rural areas. It's a fun and colorful speech pattern and it's logical enough to be explained.
First, "a man who irons" is an adjectival clause. We're not talking about men in general, we're talking about a subset of men where that subset can be defined as "men who iron". We can simplify that adjectival clause to a simple noun without changing the meaning of the "love me a <XX>" idiom, so let's do that. Let's simplify it to "coffee", to give "love me a coffee".
The word "I" is omitted, but it's understood to be there. The long form is "I love me a coffee". This is a reflexive form, similar to "I got me a coffee". But "me" is degenerate, and it should be "myself", to give "I got myself a coffee", in other words "My 'getting' of a (cup of) coffee makes it (feel like) the property of myself". So, "I love me a coffee" means "My love of a (cup of) coffee makes it (feel like) the property of myself".
Now, back to the original phrase. Hopefully you can now see that "Love me a man who irons" means "My love for a person in the set 'men who iron' makes him (feel like) the property of myself."