I know in general that stare decisis effectively means that once a court rules on a law that other courts can use that decision when considering their cases. But why is that the case, how is it applied, and why are court reluctant to go against the precedence set in another case?
The core principle of stare decisis is that the law should not depend on what judge you got; two cases with the same facts should have the same outcome. In the common-law tradition, there weren't really written statutes; there was only "what's been done in the past," and so the only reference you'd have to what the law should be in some situation is past court decisions. If judges didn't have some constraints to rule similarly to before, there really wouldn't be any legal standards (because there was no written law to go back to; in civil-law countries, there always has been a written law, so precedent isn't so important).
Stare decisis doesn't actually directly stop a judge from entering a decision that goes against binding precedent. However, judges are expected to obey precedent, and for the most part do what they're supposed to do. If they don't, the case will likely be reversed on appeal.
Binding precedent only applies within the area a court serves; a court doesn't have to listen to precedent from a different court that has nothing to do with the case. The rule is that precedent from any court up the appeals chain is binding; federal district courts are bound by their circuit court and SCOTUS, state courts are bound by higher state courts and SCOTUS (but not other federal courts, as the case can't be appealed to them). A court can sometimes overrule its own precedent, but the cases where it can do that are rare (and so applying a higher court's precedent can also be viewed as "if you appeal to them they'll say X, so we're saying X.")
Stare decisis is more than just the authority for later decisions to refer to previous ones; it is, to a certain degree, an obligation to. Which is to say, that as a rule, judges (and more often than not their clerks) will spend a lot of their time looking through old cases in order to find out what case law has decided, what the precedent is. At this point, they'll be unlikely to change it, unless circumstances have changed, or there's been a tremendous change of opinion on the court (like with the reversal in Brown v. Board of Education).
As mentioned in Auto Equity Sales v. Superior Court before the California Supreme Court and as brought in the Wikipedia article on precedent: "under the doctrine of stare decisis, all tribunals exercising inferior jurisdiction are required to follow decisions of courts exercising superior jurisdiction."
I'm not totally sure what you mean by how it's applied, but as to the reasoning: judges like having rules. We want the law to be clear and established, so that people can look through it and find out what the law is for them. If courts were constantly reversing earlier decisions, there'd be no legal stability, and it'd be far too hard for anyone to know what is and isn't the law.
This would also to a certain degree detract from the legitimacy of criminal punishment, since if the accused couldn't know it was a crime, why should we punish him?
Both of these two parts of reasoning can be found in Antonin Scalia's article on The Rule of Law as a Law of Rules (that link only contains excerpts; I wasn't able to find a publicly accessible full version of the article).
Stare decisis means let the decision stand. It use a principle used in common law systems (and is decidedly rejected in civil law systems).
If we use the U.S. Federal system as a model, there are effectively 13 court systems governed by courts of appeals (14 if we count the more limited Federal Circuit). They have responsibility for ensuring consistently among the courts below them.
The Supreme Court has the job of ensuring decisions among the 14 appellate courts are uniform.
In theory the system ensures uniformity throughout—assuming at each level, the courts follow the precedent of the court above them (stare decisis).
That's what they teach you in law school.
The reality is quite different.
First, the Supreme Court does not get to all the inconsistencies. This week the Court heard United States v. Texas. One of the issues is whether the DAPA program is an interpretive rule or a legislative rule. The Fifth Circuit (as it noted in its opinion under review) uses a standard that is different from the D.C. Circuit.
Given the nature of your question, you'd be surprised at how willing courts are to ignore precedent. The law is very chaotic at the circuit level in many areas due to conflicting, inconsistent decisions among the panels. At the district court level, judges will often thumb their nose at precedent ("Go appeal me.").
Even the Supreme Court ignores precedent to a significant degree.
In some cases, the Supreme Court will say that it is changing the law. Most of the time they don't leaving it unclear what the law is.
In addition, the Federal Court have developed the concept of the non-precedential opinion. In theory and in most cases, these are routine cases (typically prisoner appeals) where the court issues a simple opinion, citing precedent and declaring it non-precedential for efficiency purposes. However, some circuits use non-precedential opinions when they ignore precedent to get a desired outcome then declare the inconsistent opinion "non-precedential."
It only takes a few inconsistent precedents to make things unpredictable. Let's say a circuit has 200 opinions that state the law one way and 4 that state it another way. Yes, you're more likely to get the 200-opinion version of the law BUT the 4-opinion version of the law remains a possibility.
Still adding to the confusion, let's say you are in a different circuit (X) that has 300 opinions stating the law the same way as the other circuit circuit (Y) has in its 200 opinions. You might get an opinion in circuit X using the law as stated in circuit Y's 4-opinion version of the law.
The failure of the courts to follow stare decisis is the major reason we have so many lawsuits in this country. If you bring a case enough times, you can eventually get the answer you want from some judge.