At the time, it was necessary to have a large nuclear weapon capable of mass destruction because bombing was very inaccurate, and missile delivery had not yet been perfected.
The idea was that a large bomb with supermassive destruction would destroy its target, even if the actual ground-zero was hundreds of yards away. Hence, the Tsar could be dropped by bomber, and even if it didn't hit its target by ground zero, the target would be destroyed.
The Tsar, however, was not meant for actual use. Instead, it was commissioned to be a psychological weapon, with its 100 megaton yield to threaten the United States. The bomb was limited at 50 megatons, because a higher yield would not increase damage, due to the destructive force escaping into the atmosphere. Also, the increased yield would destroy the delivery aircraft.
The bomb was also impractical to use, because only 1 type of Soviet bomber, the Tupolev Bear could carry it, and it would be difficult to get the bomber, weighed down by the large bomb, into enemy airspace, drop it, and escape without being blown up by the massive shockwave. On the bomb test, the aircraft lost considerable altitude because of the bomb's shockwave.
At the time, missiles were in their infancy. However, shortly after the bomb's detonation, breakthroughs were made in missile guidance, and therefor, they could target a smaller area with greater accuracy, requiring less of a bomb yield and removing the need for a superbomb.
The tautological answer is, they didn't need it.
The explanation of that involves fleshing out a few things. One is that increasing the yield of a weapon does not increase the effect by the same amount. So the 5 psi blast radius of a 10 Mt surface burst is 6.1 mi; the same radius for a 50 Mt blast is 10.5 mi (that is, a 5X increase in yield only increases the radius by 1.7X); the same radius for a 100 Mt blast is 13.2 mi (an 100X increase in yield over 10 Mt, with only a 2.2X increase in radius). (The reason for this is because the initial blast is spherical, and so the energy gets distributed according to a cubic root.)
Or to put it another way, you can destroy about the same area with two 10 Mt bombs as you can with one 100 Mt bomb.
Secondly, let's consider what adding that yield actually does in terms of the usability of the bomb. There are maximum practical limits to how much bang you can get out of every pound of a nuclear weapon. The best the United States ever achieved was about 6 kilotons for every 1 kilogram of bomb (6 kt/kg). That was their largest yield weapon ever, the Mk-41, which managed to squeeze 25 Mt of yield (half of the tested Tsar Bomba) in a bomb that "only" weighed 10,600 lbs (just a little more than the Fat Man bomb dropped on Nagasaki).
The 100 Mt Tsar Bomba had a yield-to-weight ratio of only about 3.4 kilotons/kilogram. Which is to say, it was not a very efficient weapon. It had a big boom, but at the cost of a lot of weight. This extra weight (and size) meant that it had very, very limited delivery options.
Now, could they have done better with it, if they wanted to? Maybe. The CIA was super worried that the Soviets were trying to make 50 Mt warheads for heavy ICBMs. If the goal is to make 50 rather than 100 Mt, you can scale down the weight a lot. (The 50 Mt Tsar Bomba was really a 100 Mt design that was just decreased in yield; a pure 50 Mt variety could be lighter.) Light enough to put on a super heavy rocket? Probably — but again, there's a question of why you would do than as opposed to having just dozens and dozens of 1 Mt warheads that will do quite a lot of damage by themselves.
Could they have done it without atmospheric testing, though? I doubt it. And therein is part of the rub: the USA and the USSR signed the Limited Test Ban Treaty only two years later and that essentially halted all high-yield weapon development in both countries.
The weapon also no doubt used a ridiculous quantity of fission and fusion fuel. Which again gets to the "why would you want one" question, especially when you remember that the Tsar Bomba is a 1-bomb-to-1-bomber weapon (whereas with lower yield bombs you can fit multiple bombs in each bomber).
Sakharov, as an aside, proposed that they make a 100 Mt torpedo that could be used to destroy a harbor city like New York. Apparently the Soviet general he told this to was appalled by the idea, which Sakharov seems to have found somewhat amusing. (What, it's OK to contemplate bombing a city from the air, but not the sea?)
As a further aside, there's another, slightly more interesting question hiding here: why didn't the USA build a 100 Mt bomb? The answer is actually somewhat of an interesting story. (I gave a paper on this last November.) The short version is that there were forces within the US military/scientific/political establishment that were fairly interested in developing at least a 50 Mt bomb, just to show the Soviets (and the world) that we could do it too. Proposed delivery mechanisms at one point including having one just built into a B-52 and having the crew bail out before reaching the target. Seriously. They thought that they could even in their first generation 100 Mt bomb really out-class the Soviets in terms of yield-to-weight ratios, and that in second generation bombs (using a new H-bomb technique developed during the 1962 Operation Dominic test series) they could do much, much better. However they never went anywhere beyond the drawing board with the plans, though, because the political consequences were deemed high and because the resources involved in making such weapons were substantial (they didn't have facilities that could fabricate fissile material in the sizes and shapes necessary, for example). That and they never really developed a good idea of what the hell they would ever use them for.
The timing of the Tsar Bomba is, as just one more aside, an interesting one. The Soviets detonated it in 1961; the USA tested a series of very compact H-bombs in 1962; the LTBT was signed in 1963. The shift for the US from that point forward was mostly away from high yield weapons — the new name of the game were weapons that could fit 100 kilotons of weight into a warhead that was only 100 kg of weight (so 1 kt/kg — not the optimal efficiency, but a good weight/yield tradeoff). This meant making smaller, more accurate nukes, the sort of thing you can MIRV. Which is one of the reasons the Tsar Bomba looks like such a relic; it is a 1950s bomb, when the goal was to see how big you could make them. By the mid-1960s that no longer seemed as important as being able to have lots of bombs and to use them accurately.
For some more thoughts of mine on these topics, you might find these two blog posts useful: Kilotons per kilogram and In Search of a Bigger Boom.
It simply wasn't necessary to have bombs that large. On top of that it was impractical because it couldn't be carried by an ICBM and thus would have to be carried by a bomber which are highly likely to be shot down by ground to air missiles or by fighters.
The bombers were typically an after the fact type of deal. By the time the bombers reached their targets the war would already be long over because the ICBMs, sub launched missiles, and likely satellite based missiles would have already hit their targets, 70%+ of all electronics would already be wiped out due to the high altitude EMP bombs, and tens of millions would already be dead (edit - practically the entirety of the Soviet and United States surface navies would also be wiped out in the opening salvos by sub launched nukes). The bombers were typically just meant as a moping up effort although it really wouldn't be needed. There also would have likely been "long trigger" subs which would wait months or years after the initial war before firing their own payloads. A sort of let them start building back up and then totally fuck them again sorta deal.
There is also a point at which having a larger bomb no longer has any added effectiveness due to the curvature of the earth. After awhile the blast would no longer be at ground level but up in the air. But this point doesn't really matter when you already have the extreme impractical nature of such large bombs.
The Russian/Soviet SS-18 'Satan' missile could hold a 20-25 megaton warhead. I think that is the largest single warhead ever deployed on an actual weapon, but the modern version has up to 10 smaller warheads.