The question calls for a good answer from an Objectivist who understands Popper's work. I don't know of any Answer providers or commenters on this website (including me) who would meet that criterion, but I was able to learn at least a little about Karl Popper from a substantial article about him that I found on Wikipedia. Of all the empiricist philosophers of science whom Objectivists could study and evaluate, he appears to be a good one, well worth studying further. Perhaps Curi himself will initiate such a study and evaluation someday.
As far as I can discern, however, Karl Popper is basically an empiricist, very much in the tradition of David Hume, though not necessarily identically so. Others are welcome to comment on this point (or anything else in this Answer) if they have additional insights to offer.
Hence, my understanding is that the main Objectivist "hostility" toward Popper would simply be an expression of the same opposition that Objectivism has toward all forms of empiricism (and rationalism, too). Refer to the topic of "Rationalism vs. Empiricism" in The Ayn Rand Lexicon for a brief overview of that issue.
In fact, rather than begin by differentiating Popper from Objectivism, it might be highly fruitful to begin by comparing and contrasting Popper with Hume. One passage in the Wikipedia article on Popper (in the "Falsification/problem of induction" section) observes:
Popper and David Hume agreed that there is often a psychological belief that the sun will rise tomorrow, but both denied that there is logical justification for the supposition that it will, simply because it always has in the past. Popper writes, "I approached the problem of induction through Hume. Hume, I felt, was perfectly right in pointing out that induction cannot be logically [rationalistically?] justified." (Conjectures and Refutations, p. 55)
But Popper apparently also held that repeated unsuccessful attempts to "falsify" a theory strengthen the degree of certainty in accepting the theory as true, just as repeated confirmatory instances strengthen the degree of certainty.
If I recall correctly from David Harriman's book, The Logical Leap, a vital element strengthening a theory is the identification of a causal mechanism behind the observed results. Hume rejected causality. I'm not sure about Popper's stand on causality, though he appears (from the Wikipedia article) to have agreed with Hume, viewing causality itself as an unprovable induction. (Compare this with Objectivism's identification of causality as a corollary of identity and thus axiomatic).
I mentioned the potential value of a comparison of Popper and Hume by someone who is highly familiar with both. I would be particularly interested in Popper's stand on causality, compared to that of Hume.
I'm interested because of the Wikipedia mention of a specific work by Popper containing a specific passage allegedly in his own words, with a specific page reference, that seems to say that Popper himself saw considerable commonality with Hume regarding induction. I also know from other sources that Hume rejected causality, which is essential (according to Objectivism) to valid induction. My previous Answer above indicates the Objectivist view of what causality refers to and why it is valid. (Ayn Rand identified causality as the law of identity applied to action.) I also see from the comments that Wikipedia is accused to getting Popper completely wrong. So naturally I'm interested in hearing more about this from someone who knows more about it.