This is the study the article mentions.

Does the study have those conclusions?

Not really. The experimental results could equally be explained by a simpler conclusion, that people are less likely to ask others out when they feel disappointed, deflated or embarrassed by something (in this case, the public revelation of a poor test result). The researchers acknowledge that such an explanation is potentially valid and supported by prior research on "affective experience", and acknowledge this as a limitation.

It's also just one study, on about 100 U.S. psychology undergraduates, with a systemic bias excluding more intelligent students. It therefore can't be generalized without being replicated on other populations.

Here's an example of the kind of target used by the study:

Seventy-three male undergraduates (M age = 18.95) participated in a “Study of Interpersonal Attitudes” in exchange for psychology course credit.

Age, field of study and personal maturity are confounding factors on psychological behavior. As such, the news reporting by "The Independent" is incorrect and misleading.

Is the study reputable/reliable?

While the study is "serious science", is being peer-reviewed (or is in pre-print) and is quite interesting, the methodology has systematic bias against intelligent men, and the conclusions are not supported by the experiment set-up: the study merely shows that embarrassing or diminishing people in front of a potential date makes them feel a bit more antagonistic towards her. Likewise, it shows that flattering people in front of a potential date makes them feel more sympathetic and likely to ask them out. As reported, it does not measure anything else.

Systematic bias

The study in question consists in letting participants take a math test, and then be told that their result was average.

After a few minutes, the experimenter returned with the “graded” tests and announced their scores while handing back the tests. Participants always got 12/20 questions correct

At the end of the experiment, they were asked whether they believed the results and were discarded if they did not. This is a clear systematic bias because it excludes all candidates that are smart enough to expect a nearly perfect score and know that they can't possibly have an average result.

[P]articipants responded to the following questions in written form: “What do you think this experiment was about? Was there anything odd or suspicious about this experiment? If yes, what?” Responses were later coded for whether participants mentioned (a) that their partner was not real or part of the study or (b) thought their test score was fake. Participants who reported suspicion on one or more of these items were excluded from analyses.

In fact, it excluded 10% of the participants.

Participants and Procedure

Ninety male undergraduates participated in a “Study of Interpersonal Attitudes” in exchange for psychology course credit. Nine participants were excluded because they were suspicious of the confederate or did not believe the test feedback; the final sample consisted of 81 participants (M age = 18.81).

Validity

First an experiment is set up where the fake results are given in a separate room from the potential date. In this case, performing worse than the date predicts more likelihood to ask the partner out or find them attractive.

Secondly another experiment is set up so that the fake results are given in front of the potential date. In this case, performing better than the date predicts more likelihood to ask the partner out or find them attractive.

I don't see how that supports that "men are threatened by intelligent women". It simply shows the obvious result that when someone's ego is boosted by good results they are more optimistic and when their ego is deflated they become timid.

The authors themselves admit to that, although they simply dismiss this as a "limitation".