I work in the subject, and I think I can be of help explaining this.
I'll explain it using the water analogy:
Electric current flow -> Water flow
Voltage -> pressure
If you have a network with nodes, and branches; the nodes are where the water is injected and subtracted from the network, and the branches are the pipes.
(In electrical grids, the pipes are transformers and lines, while the nodes are the nodes or busbars)
If you have "water" injection in a node that originally was designed for consumption, then the pressure in the pipes might increase up to a level where the pipes break.
(This would be solar production at household level)
The same way, too much consumption at a node might lower the pipes pressure too much and the system will not work.
The way of dealing with this is to store the surplus of energy and supply it when needed, that is why batteries are the holly grail of renewables.
Huge renewable penetration is a situation that grid operators and electrical companies are against because it forces them to adopt new approaches to a job they've been doing for a century with few radical changes like the ones they need to make. (My opinion)
I hope this is clear enough, otherwise I can explain things further since this is my daily work.
[EDIT: Why do the pipes break?]
Well as you requested, I'll go a bit more in detail here:
Each branch element (lines & transformers) has a limit in the amount of current can go though it without overheating and set on fire. This nominal current can be overpassed for a limited amount of time, so an overload is not a life or death event, if it does not last too long (Also overloads diminish the elements life)
On the other hand, the voltage should be within a +-5% of the nominal voltage of a node, this is 230V +-5% per phase (In Europe, in US is 125?). Generating power in a node increases the voltage in that node and in the neighbour nodes (For the same load situation) Increment in demand in a node decreases the voltage in that node and its neighbours). This is why if I put a massive amount of solar panels at home I might get voltage issues at my house and at my neighbours houses. This issue can be mitigated by proper inverter firmware programming, but there is no regulation on that in many countries, so there are this problems people haven't heard about but are very real.
But why does the voltage has to be in such a limit?
Well this limits is a security constraint set by grid operators. If the voltage in the sockets of your house is too high it might break the power electronics of your devices (PC's TV, etc..) if the voltage is is too low, electronic devices might not work or even break as well. An incandescent light bulb shines brighter on high voltage, and less brighter on lower voltage.
Tell me if more details are needed.