Being an equal opportunity employer means that the institution abides by the US federal law on Equal Opportunity - basically that the employer cannot consider race, color, religion, sex, or national origin when making personnel decisions. Additional protected classes, such as age, have been added to Equal Opportunity over the years. Equal Opportunity does not force an institution to hire someone they would rather not hire on the basis of being a member of a protected class, but it means they cannot artificially exclude that candidate. Equal opportunity means that all applications must be evaluated fairly. It does not mean that the best candidate on paper always gets the job.
This is in contrast to Affirmative Action, which in the US is "positive discrimination". In other words, the institution is forced to artificially maintain diversity in its staffing if it does not happen naturally. Affirmative action can mean that you would have to hire your second (or third) choice over your first choice to maintain diversity.
Edit - to address some comments.
From the perspective of an applicant, EO violations can difficult to prove, and the burden of proof is usually on the applicant. Most of the information collected during a hiring process is not (and never will become) public. It is shredded/deleted once the position is filled. An applicant's on paper qualifications are nearly impossible to know, unless you are a close acquaintance of the person. The CV that person posts on their website might be abbreviated, incomplete, or out-of-date.
Additionally, most hiring committees often consider important intangibles that are assessed during the interview. Academic positions are different from positions elsewhere. Most academic committees are assuming that they are hiring a person who will stay at that institution, get tenure, etc. They need to assess if the candidate is likely to do so, if the candidate is likely to thrive in the current culture of the department, if the candidate will get along with the faculty personalities, if the candidate is likely to start looking for something bigger and better five-ten years later, if the candidate is likely to survive the tenure process, if the candidate's specialty/expertise fill a current void in the department, if the candidate is likely to meet service expectations with enthusiasm, if the candidate is amenable to the crappy teaching load he or she will get the first semester, if the candidate's research plans are feasible given other expectations and institutional resources, etc.
You may have more awards, publications, conference presentations, fellowships, grants, letters after you name, or whatever on-paper metric you are using, but you may not have met the needs of the department or institution as well as the other candidate. You may feel more qualified, but someone else was a better fit overall for the position. This question and the answers to this question suggest that applications that look too good may not be considered seriously because hiring committees are afraid of a bad fit.
Alternatively, as may happen, the department you applied to may have always intended to hire the person they hired, but to meet legal and institutional requirements they conducted a search. They interviewed several other strong candidates from diverse backgrounds to satisfy EEO requirements, and then they hired the person they wanted to hire. I won't pretend that such behavior isn't shady, but it can be common, especially for administrative positions, and it is legal. The best way for a hiring committee to prove they are following EEO guidelines is to interview a diverse array of qualified candidates.
To allege that your application was artificially rejected because of your membership in a protected class, start by getting a lawyer. Don't risk jeopardizing your case by inadvertently proceeding without one. Then, do whatever your legal counsel advises. That is what they are for. The only real proof that the hiring committee discriminated against you is if no members of your protected class were interviewed. If you were interviewed, that means they either 1) considered you a strong candidate or 2) went out of their way not to exclude you. Either way, you have almost no case. It may make you angry if the second thing happened, but there is little you can do except to warn people you know away from applying at that institution. If they discriminate against you because you are not the one person they really want for the job, it is not illegal, it just sucks.