Why charging Li-Ion batteries in cold temperatures would harm them?
'Cold temperatures' is awfully vague.  First, let me actually specify some real, hard numbers.

Do not charge lithium ion batteries below 32°F/0°C. In other words, never charge a lithium ion battery that is below freezing.

Doing so even once will result in a sudden, severe, and permanent capacity loss on the order of several dozen percent or more, as well a similar and also permanent increase in internal resistance.  This damage occurs after just one isolated 'cold charging' event, and is proportional to the speed at which the cell is charged.  

But, even more importantly, a lithium ion cell that has been cold charged is NOT safe and must be safely recycled or otherwise discarded.  By not safe, I mean it will work fine until it randomly explodes due to mechanical vibration, mechanical shock, or just reaching a high enough state of charge.  

Now, to actually answer your question: why is this?

This requires a quick summary of how lithium ion batteries work.  They have an anode and cathode and electrolyte just like any other battery, but there is a twist: lithium ions actually move from the cathode to anode during charging and intercalate into it.  The gist of intercalation is that molecules or ions (lithium ions in this case) are crammed in between the molecular gaps of some material's lattice.  

During discharging, the lithium ions leave the anode and return to the cathode, and likewise intercalate into the cathode.  So both the cathode and anode act as sort of a 'sponge' for lithium ions.  

When most of the lithium ions are intercalated into the cathode (meaning the battery is in a fairly discharged state), the cathode material will expand slightly due to volumetric strain (because of all the extra atoms wedged in between its lattice), but generally most of this is intercalation force is converted to internal stresses (analogous to tempered glass), so the volumetric strain is slight.  

During charging, the lithium ions leave the cathode and intercalate into the graphite anode.  Graphite has is basically a carbon biscuit, made of a bunch of graphene layers to form an aggregate biscuit structure.  American biscuit structure.

This greatly reduces the graphite anode's ability to convert the force from the intercalation into internal stresses, so the anode undergoes significantly more volumetric strain - so much so that it will actually increase in volume by 10-20%.  This must be (and is - except in the case of a certain Samsung phone battery anyway) allowed for when designing a lithium ion cell - otherwise the anode can slowly weaken or even ultimately puncture the internal membrane that separates the anode from the cathode, causing a dead short inside the cell.  But only once a bunch of joules has been shoved into the cell (thus expanding the anode).  

Ok ok, but what does any of this have to do with cold temperatures?

When you charge a lithium ion cell in below freezing temperatures, most of the lithium ions fail to intercalate into the graphite anode.  Instead, they plate the anode with metallic lithium, just like electroplating an anode coin with a cathode precious metal.  So charging will electroplate the anode with lithium rather than, well, recharging it.  Some of the ions to intercalate into the anode, and some of the atoms in the metal plating will intercalate later over 20+ hours if the cell is allowed to rest, but most will not.  That is the source of the capacity reduction, increased internal resistance, and also the danger.

If you've read my related answer on stack exchange to the question 'Why is there so much fear surrounding lithium ion batteries?', you can probably see where this is going.  

This lithium plating of the anode isn't nice and smooth and even - it forms in dendrites, little sharp tendrils of lithium metal growing on the anode.

As with the other failure mechanisms which likewise are due to metallic lithium plating of the anode (though for different reasons), these dendrites can put unexpected pressure on the separating membrane as the anode expands and forces them into it, and if you're unlucky, this will cause the membrane to one day fail unexpectedly (or also immediately, sometimes a dendrite just pokes a hole in it and touches the cathode).  This of course makes the cell vent, ignite its flammable electrolyte, and ruin your weekend (at best).

But, you might be wondering, "why do below-freezing temperatures cause lithium metal plating of the anode?"

And the unfortunate and unsatisfying answer is that we don't actually know.  We must use neutron imaging to look inside functioning lithium ion cells, and considering there are only around ~30 (31 I think?) worldwide active research reactors (nuclear reactors that act as a neutron source) that are actually available for scientific research at a university rather than used for medical isotope production, and all of them booked 24/7 for experiments, I think it is just a matter of patience.  There have only been a few instances of neutron imaging of lithium ion batteries simply due to scarcity of equipment time.  

The last time this was used specifically for this cold temperature problem was 2014 I believe, and here is the article. 

Despite the headline, they still haven't really solved exactly what it is that causes plating rather than intercalation when the cell is below freezing.

Interestingly, it is actually possible to charge a lithium ion cell below freezing, but only at exceedingly low currents, below 0.02C (so more than a 50 hour charge time).  There are also a few exotic cells commercially available that are specifically designed to be chargeable in cold temperatures, usually at significant cost (both monetarily and in terms of the cells' performance in other areas).  

Note: I should add that discharging a lithium ion battery in below freezing temperatures is perfectly safe.  Most cells have discharge temperature ratings of -20°C or even colder.  Only charging a 'frozen' cell need be avoided.