How things are like? vs What things are like

The use of how things are like is highly unusual, although not actually ungrammatical.

I find it doubtful that anybody would actually mean to use how in that sentence. Most likely, what they actually mean should be expressed with what instead:

"I don't know what things are like nowadays."
"Why, they're crazy."

If this is what's meant, then using how instead of what is a mistake.

However, if the use of how is deliberate, then it's both grammatical and actually sensible:

"I don't know how things are like nowadays."
"Why, the whole world's behaving like a sick person."

In other words, both how and like are used in their normal sense, allowing the original sentence to be rephrased as follows:

"I can't think of an analogy for the manner in which things are functioning these days."

So, you can say the sentence in the question, using how rather than what, and have it make sense. However, it's unlikely for the person saying it to actually mean it in the sense that allows how to be a reasonable word choice.

If somebody actually does want to express the how meaning, it would be better for them to rephrase the sentence so there is no confusion on the part of anybody hearing it.