Based on this:
May wanted to push her language skills, so before joining Davis Polk, she did an intensive Mandarin study in Beijing for two months.
what does "push her language skills" mean?
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The connotation of "push" in this context is that May wants to challenge herself. If you read further down in the paragraph from the article you linked, you see May quoted as saying, "Davis Polk is a great place to be challenged." The author of the sentence you quote is probably trying to find a way to reinforce this theme of "being challenged" that May brought up in her interview without using the same word repeatedly.
The way this sentence is framed--"push [a skill]"--is a bit unusual and may sound awkward to the English-speaking ear. You will more commonly see "push [him/her/my/yourself]" or "push [a person]", e.g.:
"May had taken Mandarin classes in America for several years, but she really wanted to push herself, so she traveled to Beijing for a language-immersion experience."
"The runner was exhausted, but he pushed himself to complete the last few miles of the marathon."
The teacher pushed his students to do their best work.
"Push" may also carry the connotation of "force," even when physical pushing is not involved, depending on the context, e.g.:
The mine shaft had failed a safety inspection, but the supervisor pushed the miners to enter it by threatening to dock their pay.
I think they're borrowing the meaning from the idiom "push the envelope" which means "to expand the definition, categorization, dimensions, or perimeters of something".
By saying "push her language skills", they mean to "expand the dimensions and perimeters or her language skills", i.e. make it larger. Augment it.
to follow up vigorously; promote (a campaign, claim, etc.)
to extend or expand (business activities, etc.)
So in this example of yours, it means to increase her skills.