How do chemical reactions taking place in an alkaline battery produce electrons?
An alkaline battery is a specific type of galvanic cell.  As I am sure you found in your research, it involves an oxidation-reduction reaction between zinc($\ce{Zn}$), and manganese(IV) oxide ($\ce{MnO2}$.)  They are called alkaline batteries because these chemicals are surrounded by an alkaline (basic) solution of potassium hydroxide, $\ce{KOH}$.  

Your question is really one that is more about galvanic cells in general.  When an oxidation-reduction (a.k.a. redox) reaction occurs, electrons are transferred between species.  This sort of reaction is exceedingly common.  That's all that happens in a battery, it's just that people have figured out how to catch the electrons on their way from species A to species B and use their energy to do work.

If I took a piece of zinc and placed it in a slurry of $\ce{MnO2}$ and $\ce{KOH}$, it would react according to the following balanced chemical equation and produce a lot of heat.

$\ce{Zn + 2MnO2 &lt;-&gt; ZnO + Mn2O3}$

Suppose that I separated the two pieces of the reaction into their own jars.  A piece of zinc placed in a solution of $\ce{KOH}$ in one jar and some $\ce{MnO2}$ mixed with $\ce{KOH}$ in another.  The reactants are all present, but they have no way of interacting, so nothing would happen - there's no way for electrons to move from the zinc to the $\ce{MnO2}$.  

Now I attach a piece of wire to the zinc on one end and into the $\ce{MnO2 - KOH}$ slurry at the other. Now electrons can move from one jar to the other.  If I have a voltmeter that can measure a very brief signal, I can actually see a quick pulse of electrons flowing through the wire, from the zinc to the $\ce{MnO2}$.  The pulse ends almost immediately though, because the movement of electrons away from the zinc and toward the $\ce{MnO2}$ creates a charge separation.  The former becomes positively charged and the latter negatively charged.  Continuing to move electrons against this charge separation is progressively more difficult, and it doesn't take long for the electrostatic force of the charge separation to balance out the electrochemical potential of the reaction and the entire thing just stops.

To fix this problem, I need a way for the charges to balance back out without allowing the reactants to mix directly.  This is done with some sort of salt bridge.  A salt bridge allows positive and negative ions to move between the jars, but it still keeps the jars separate from one another.  This could be as fancy as an actual salt bridge (a glass tube filled with a gel containing some sort of salt) or as simple as a piece of paper soaked in the $\ce{KOH}$ solution.  Now when I connect the wire, electrons flow from the zinc to the $\ce{MnO2}$, negatively charged $\ce{OH-}$ ions move through the salt bridge toward the zinc, and positively charged $\ce{K+}$ ions move through the salt bridge toward the $\ce{MnO2}$.  The charge stays balanced in both jars, so there is no electrostatic force preventing electrons from moving, and they flow freely through the wire.  If I put a motor in the middle of the wire, it turns.

Specifics of Alkaline Batteries

That all was regarding galvanic cells in general.  A battery has additional requirements.  It needs to be compact and safe, in particular.  The construction of an alkaline battery involves a zinc core with a paste of manganese(IV) oxide and potassium hydroxide wrapped around it.  The paste and zinc are separated by a paper sheet (the salt bridge in this galvanic cell.)  Because all of the reactants and products are solid, there is no concern that they will diffuse through the cardboard; it just keeps the reactants in their places.  

The reaction does not start until the battery is put into a device and the device is turned on.  Flipping that switch gives the electrons from the zinc a pathway to reach the manganese(IV) oxide.  Until the circuit is complete, there is no buildup of electrons at one end of the battery.

Electrical potential (voltage) is the ratio of the energy carried by an object to that object's charge.  One electron in this system has a potential of 1.43V, two electrons carry twice as much energy, but they also have twice the charge, so they also have a potential of 1.43V.  Electrical potential is sometimes called electromotive force, and that idea, that we're measuring how hard the electrons are being pushed through the wire, may help you understand the battery's behavior better.

Why 1.43V though?  It's because of the properties of the chemical reactions the battery is based on.  You probably know that if you burn some amount of propane it will produce a given amount of heat, and if you burn an equal amount of acetylene, it will produce a different amount of heat.  The same thing is true here.  The energy associated with the half reactions in each part of the battery depends on the chemicals involved.  Rather than list a total amount of energy available, batteries list the energy per charge, voltage.  This battery's half reactions are:

$\ce{Zn + 2OH- -&gt; ZnO + H2O + 2e-}$  E°=1.28V
and
$\ce{2MnO2 + H2O + 2e- -&gt; Mn2O3 + 2OH-}$ E°=0.15V

Each half reaction gives a specific amount of energy to the electrons involved in it.  Since the energy:charge ratio (E° above, called a reduction or oxidation potential) is constant, we can use that to find the energy:charge ratio of the entire battery just by adding them together.  Why doesn't it go above 1.43V?  For the same reason that a fire doesn't burn hotter.  The chemicals involved only release so much energy per reaction.  Note: The potential can be changed in some systems by playing around with the concentrations of the chemicals in the cell.

As the battery is used up, the zinc slowly but surely turns into zinc oxide and the manganese(IV) oxide turns into manganese(III) oxide.  As their amounts fall to zero, the battery's potential falls and it quits working.