One "hot" wire be pigtailed into multiple "hot" wires. How does that affect the electricity going through them?
Analogies can get you in trouble when you extend them past the point they are useful.  I will present a different one that is more applicable to your situation.

Electricity in your house is basically a pull system.  If I plug (turn on) an electrical device, it will attempt to draw its rated power.  A 60 W light bulb will draw 60 W, a 1000 W hairdryer will draw 1000 W, a small TV may draw 200 W.  For a North American system as this appears to be, the power is supplied inside your home at 120 V.  We convert Wattage into current using I = W/V.  So the light bulb will draw 0.5 A, hairdryer, 8.3 A and TV 1.6 A.

The wiring you show appears to 14 AWG which is typical household wiring and is rated at 15 A in most applications.  I would expect a circuit breaker is protecting the wire and it will trip above 15 A.

The single circuit into the pigtail will be able to handle a maximum of 15 A.  The two downstream pigtail circuits will provide as much current as needed by your loads (like the light bulb, hair dryer, or TV) up to a combined 15 A.  Some more math here, in = out1 + out2.  So if it is 15 A, either pigtail can handle up to 15 A, one at a time but not simultaneously.  Or your can have any value where the two pigtails add up to 15 A.

So a pigtail with multiple outputs can supply multiple loads without any detrimental effects as long as the combined load can be supplied by the input wire and circuit breaker.

Additional details to be ignored:

I have ignored many factors here that are not relevant to this question.  Sustained current may be limited to 10-12 A not 15 A.  There will be voltage drops as the load increases.  My 60 W equivalent LED is much lower actual wattage than the older light bulbs.  I have assumed the correct breaker is installed and wiring is to code.