You have essentially answered your own question, at the most basic level, overstretches supply line meant that it was in efficient or impossible to adequately resupply the frontline.

Now this could mean a number of things.

The first is that the means to deceiver supplies to the frontline were either inadequate or non-existent. In the context of the First World War, this was usually the case. As an attacking force advanced, usually across No Man’s Land which was pockmarked with shell holes, barbed wire and the detritus of war, it became harder to resupply them as supplies usually had to be carried by men following behind. This was slow and it was dangerous and it required a large number of men to carry enough supplies to keep the attacking force resupplied. This was one of the major obstacles that both sides faced during the war as they attempted to advance. As the war went on, the situation only somewhat improved with the development of narrow gauge light railways which could deliver large numbers of supplies to the rear areas but could not go into the front lines. Motor vehicles couldn’t traverse broken terrain and were restricted to the exisiting and surging road networks. They did experiment with supply tanks which proved to be somewhat successful and they also experimented with dropping ammunition from aircraft to advancing troops. The problem in the First World War was never truly solved though as communication with frontline forces was always an issue making locating a unit and determining what they needed was incredibly difficult.

During the Second World War, motorisation made supplying armies (especially Allied armies which were by far the most motorised armies during the war) far easier. Germany had greater difficulties as it still relied heavily on horse drawn transport as well as a restricted access to oil. The US and British advance eastwards towards the German border in the latter half of 1944 is an excellent example of overstretched supply lines. In this case, it wasn’t an issue of getting supplies to the front, the road networks could for the most part support Allied logistics, the issue was that their simply wasn’t enough of everything to go around, especially fuel. During the US Third Army’s advance towards the German border, General George S. Patton was repeatedly frustrated by the lack of supplies being allocated to him. This was because General Eisenhower needed to spread the supplies out along the entire front so that the frontline moved broadly as one rather then a single army driving deeply into German lines. Patton was frustrated even further when most of his fuel was allocated to the British for Operation Market Garden, the attempt to breach the German border via The Netherlands. In this case, it want an issue of getting the supplies to the forces, but rather a lack of adequate supplies for the forces that needed them.

The biggest factor is ammunition. It will deplete rapidly in military operations of the magnitude we saw in WW1 and WW2. Units without will panic, fold and retreat rapidly and even if it did fight, it would be completely ineffective. You need either fresh troops or supplies to continue fighting.

And to get either of those you need a working supply system. The examples I'll give are most for the European theater and mainly western Europe in the first world war.

A non-active infantry division of 10.000 to 18.000 men might only need 45-80 tonnes of goods daily but this could easily rise to more then a 1000 tonnes in heavy fighting. The biggest increase is off course ammunition and, more so during WW2, fuel.

During both wars the main way to supply the troops was by train which would be a 100 wagons per infantry division to get the goods to the supply depots. In both world wars horses were vital to get the goods to the divisions itself. These would be replaced by trucks more and more but it's important to know horses remained vital for nearly every army until the end of WW2. The only exception was the USA in the second world war.

So the ammunition comes by train from the factories to warehouses, from there to go by train to army supply dumps, then horses and trucks take these to the division supply dumps from where they get transported again to the battalions who supply it to the companies who are fighting. Every step of the way can cause it's own problems.

The start is the manufacturing itself and here the supply chain can fail on a macro level.

You not only need the factories but a massive workforce, tons of materials and chemists to streamline processes or search for alternatives. One of the reasons the Germans eventually collapsed in both wars was because they didn't have the materials to sustain their war industry.

https://ichef.bbci.co.uk/news/624/media/images/58531000/gif/_58531439_explosive_prod_ww1_304gr.gif

The next step by train is from the depots to the respective army supply dumps. This can pose all kinds of different problems.

This can be over a different rail network in a foreign country like Soviet Russia who had wider tracks then Nazi German railways. The Germans had to change the tracks to be able to use trains.

Sabotage by the invaded country like Belgium who disabled their railway network during WW1 to great success. Only 15% or so was operational even after one month of intensive repairs by the Germans. It's one of the reasons the Imperial Army didn't manage to reach Paris.

Air control, during WW2 the USA and UK had almost complete air control over western Europe from 1944 onward. Germans could only move armies and supplies during the night greatly hindering its effectiveness.

From there it had to pass to it's respective divisions, then to the battalions and finally to the fighting men. Like mentioned previously this was done by horse and truck and to limit the strain divisions could only operate so far away from a train station.

A lack of roads, bridges etc combined with sabotage, partisan ambushes or air raids make things even more complicated. A prime example of a failure in the last "chain" of the supply chain is the western front during WW1.

The continuous trench warfare had completely destroyed the landscape and it's infrastructure and although tactical success could be achieved on the field of battle it was nearly impossible to shift the supply lines far enough to support the fighting troops through the hellish lands.

TLDR: Men in combat start shooting a lot, really a lot. When they can't shoot anymore any offense is suicidal and nearly any defense will fold like paper. If bullets and shells don't get made and don't end up at the front your supply chains have failed.