How can net force equal to friction? I am struggling to understand a concept in one of my problems:

A $65\,\rm kg$ ice skater coasts with no effort for $75\,\rm m$ until she stops. If the coefficient of kinetic friction between her skates and the ice is $\mu_\mathrm{k} = 0.10$, how fast was she moving at the start of her coast?

Now I was drawing my free body diagram and I determined there are two forces acting on this person: 
$F_\mathrm{initial} - F_\mathrm{friction} = F_\mathrm{net}$
Because since there is a starting push, that force would be acting on the person all through the $75\,\rm m$ until the friction and force are the same magnitude, stopping the movement.
However, I was told that the net force equals the kinetic friction of the person. How can this person only have friction force?
 A: I see three crucial misunderstandings that I would like to point out:
Correction 1


*

*A spaceship is moving with engines off, drifting through space. There are no forces.

*A spaceship is staying still, hanging in space. No speed. Also no forces. 


Conclusion? Forces have got nothing to do with speed. No forces (no net force) means no change in speed. Not no speed. Your statement:

until the friction and force are the same magnitude, stopping the movement

is false. Balancing forces won't stop a motion.
This is from Newton's 2nd law:
$$\sum F=ma$$
A net force doesn't cause speed, it causes changes in speed (acceleration). No net force causes no change in speed.
Correction 2


*

*A spaceship is hit by a comet. A wing breaks off from enormous impact force. The wing flies away. But doesn't speed up. Nothing slows it down (no friction in space) - and nothing speeds it up. It drifts off at constant speed forever until it reaches and interacts with somethings new.


Conclusion? The force that gave it the speed, doesn't work on it anymore. Your statement:

Because since there is a starting push, that force would be acting on the person all through the 75m 

is false. A force only works while being exerted by something. As soon as the cause of the force disappears, the force disappears.
This also answers your title question:

How can net force equal to friction?

It can, since the other force you thought was there isn't there. The force diagram should horizontally show only friction. 
Correction 3
Kinetic friction follows Amontons law in everyday cases:
$$f_k=\mu_kn$$
No mention of speed, distance, time or similar here. Experiments show that friction is proportional to normal force, only. It will not grow or reduce along the way. Kinetic friction is constant while the sliding takes place.
Your statement:

until the friction and force are the same magnitude

is impossible since none of those two forces change during the motion. They never grow or reduce to become equal.
A: The starting push, results in a starting velocity (which is asked for). As such, it has an influence on what follows, but that starting push is not a force that acts on the skater for the next 75 m
A: The weight of the skater is equal to the normal force that the ice produces.  This allows calculation of the friction force that is slowing the skater down.  The friction force does work over the 75 m that she skates as she comes to a stop.  In addition, the work/kinetic-energy theorem allows you to calculate her starting kinetic energy (which is equal to the work done by friction), and hence, her starting velocity.
