My understanding is that in the decision for Marbury v. Madison, the Supreme Court asserted its authority in determining if laws were or were not Constitutional, and this decision is the basis for how SCOTUS operates as we know it. But if SCOTUS had to assert this authority in a case, that authority must not have been explicitly granted in the Constitution - it feels like SCOTUS "bootstrapped" its power to check and balance the executive and legislative branches. How did the other branches of government react to SCOTUS "giving itself" power not expressly enumerated in the Constitution?
Before the Marbury v. Madison case, the Supreme Court had not made any decisions that would significantly affect United States law, and was hardly recognized as an equal power to the other two branches. It had been realized that the courts had this power, but it was not until the SC had to step into the Marbury v. Madison that people had seen the power that they possesed.
Marbury tends to get simplified a bit when discussing the nuanced effects of what exactly happened in that case.
As background, pretty much everyone agreed that the judiciary had the job of resolving disputes by interpreting the law. And sometimes the law has more than one source.
So there's a rule out there, that came up in the common law, that verbal contracts are binding contracts. But the English parliament passed a statute called the Statute of Frauds, modifying that rule saying that contracts for transfers of interests in land, marriage contracts, any contract with more than a year's duration, and a few other types of contracts had to be in writing. So courts dealing with these issues had a parliamentary rule grafted onto their existing common law rules.
And as you can imagine, Parliament (or its closest American successor, the U.S. Congress) can pass new statutes repealing or modifying previous laws. So courts would sometimes have to interpret inconsistent statutes to resolve a conflict.
Marbury's main innovation is to say that it would not defer to Congress (who wrote the statute) or to the President (who signed the bill) in determining whether a statute passed by Congress was consistent with the meaning of the Constitution, and in clarifying that the Constitution trumps any statute passed by Congress.
But think for a second whether an opposite result would even be possible. Let's say Congress passes an ex post facto law in 2016 saying that anyone who mined bitcoins in 2015 retroactively committed a crime. Then what happens? The executive branch's prosecutors can bring criminal cases against 2015's bitcoin miners, but the court that imposes a sentence must agree that the criminal law is valid in order for that law to have any effect. It's not "bootstrapping" for the courts to say "we're not going along with the other two branches' interpretation of the Constitution." So in a way, the independence of the judiciary (through constitutional design) makes judicial review inevitable. Laws that don't need to be interpreted by the courts can't be struck down by the courts, either (due to the way we've set up the requirement that federal courts only hear live cases or controversies). Laws that purely govern Congress or the executive branch tend not to make it into court cases, and the political question doctrine prevents the courts from interpreting many of the ones that do come up.
So Marbury isn't truly a bootstrapping problem. It's that the courts have told the public and the other two branches that it would not exercise its judicial power in a way that it doesn't independently believe to be constitutional.
Edit: it's hard to condense this into sources from my phone, but a great starting point is Akhil Amar's Marbury, Section 13, and the Original Jurisdiction of the Supreme Court. It's mainly about another issue, but its background section makes the argument that people overstate things when they say that Marbury gave the Supreme Court the power to declare laws unconstitutional.