Non-Political Description
In favor of legal abortion services and legal access to them.

Political Description
Receiving or performing abortion services as a matter of personal choice and privacy. Supporters of the pro-choice agenda do not necessarily support abortion itself, only the position that women are entitled to make the decision themselves.

Political Benefits
Infers that those who are not “Pro-Choice” are by definition “Anti-Choice.”
Invokes the idea that personal freedom and liberty is under threat if one is not “Pro-Choice.”
Avoids referencing the word abortion which is the concept at the heart of the issue.

Reference Point
Self and others who share same political view. Unsurprisingly, this euphemism is not used by Pro-Choice opponents who instead refer to Pro-Choice advocates as Pro-Abortionists.

History
I was unable to confirm the origins of the term. What follows is the best I found.

The organization known as NARAL, National Abortion Rights Action League, was founded in 1969 by Betty Friedan and others as the National Association for the Repeal of Abortion Laws, and described themselves as pro-choice.

This term surfaced in sloganeering to avoid the harness of ‘pro-abortion.’ This inspired use of ‘choice’ reflected polls that showed more people identifying themselves as being ‘in favor of a woman’s right to choose’ than ‘in favor of legal abortion.’ Alan L. Otten of the ‘Wall Street Journal’ first used the term in print.” That was in a March 20, 1975 article.

Example Usage
From the Philadelphia Inquirer:

City Council yesterday rescinded its week-old proclamation that Philadelphia is a “pro-choice city,” putting an end to a purely symbolic debate that drew the attention of Cardinal Justin Rigali and roiled Council as few other issues had in months…Sponsored by Democratic Councilwoman Blondell Reynolds Brown, the resolution stated that Philadelphia supports “women’s reproductive rights and freedom” and defends “the right to choose a legal and safe abortion as a final but critical option for women.”