There's several sides of the story here: (1) ventilation, (2) oxygenation, (3) hemoglobin-oxygen dissociation curve
Julia did a great job explaining. I'll go into more detail as it relates to a physiology. It's imperative to understand all the players in the game and to keep each one in mind in the clinical setting, so that nurses, physicians, or other clinicians can take action and intervene appropriately and therapeutically.
VENTILATION:
First, we'll cover ventilation. You and I initial breaths via negative pressure ventilation (NPV). Our diaphragm contracts downward and our thoracic cage "expands" which creates a "negative pressure" relative to the atmosphere. There's other muscles involved, but we'll stick with just the diaphragm to keep it manageable. This negative pressure "sucks in" atmospheric air.
This atmospheric air travels through our conducting airways down to our terminal, small, and respiratory airways and participates in gas exchange. With asthma your airways "get clogged up" (mucus) and "narrow" (bronchoconstriction). As you can imagine, this will greatly diminish your ability to ventilate. So, problem one starts with the inability to ventilate effectively. You cannot get air in, and you cannot get air out.
Oxygenation
It is possible to oxygenate without ventilation (i.e. insufflation or CPAP with 100% oxygen to a non-dependent lung during one-lung ventilation). While oxygenation and ventilation are often used interchangeably, they are not the same thing. There's a lot of other factors that play into this like the partial pressure of alveolar oxygen, partial pressure of arteriolar oxygen, dead space vs shunt (V/Q ratios), diffusing capacities, hemoglobin, blah blah blah....
But to drive the point home, oxygenation is not the same as ventilation. One can ventilate all he/she wants, but if he/she is breathing a hypoxic mixture (100% Nitrous Oxide, for example), he/she is not going to be oxygenating. And if he/she is not oxygenating, he/she will not be ventilating for very long either...
Building on these concepts as they relate to asthma, you can see how oxygenation would be affected by the inability to ventilate effectively.
Oxygen-Hemoglobin Dissociation Curve
Hemoglobin, as you know, hangs out our red blood cells and has a special affinity for both oxygen and CO2. For this question, we'll assume that the asthmatic patient has "healthy, adult hemoglobin" and is not affected by pathologic conditions. Hemoglobin's affinity for oxygen is not constant, though. There are many things that affect hemoglobin's affinity for oxygen. But we'll stick to the oxygen and CO2 relationship as it relates to hemoglobin.
In short, our bodies are wonderfully engineered to "receive" or "load" oxygen onto the hemoglobin when and where we need it most (i.e. in the lungs). The inverse is also true, hemoglobin "unloads" or "deposits" oxygen where our bodies need it most -- in our tissues, etc. The way hemoglobin "knows" to do this is its special relationship with both oxygen and CO2.
As the partial pressure of CO2 is rises, the oxygen wants to "unload" (this is in the tissues); and as the partial pressure of CO2 lowers, the oxygen wants to "load" onto the hemoglobin (this is in the lungs). You can look these up further it you'd like: the Bohr Effect & the Haldane Effect. In essence, as CO2 increases, this will shift the oxygen-hemoglobin dissociation curve to the right, which means that oxygen will want to UNLOAD / leave hemoglobin. As a rule of thumb, any kind of acidosis will have this affect on hemoglobin's affinity for O2. The inverse is also true, alkalosis will generally cause a left shift and will increase hemoglobin's affinity for oxygen -- the oxygen will be less likely to "leave" the hemoglobin.
So, as the asthmatic is struggling with his/her obstructive airway issue and the diminished ability to ventilate you will have the following:
(1) Diminished gas exchange & oxygenation
(2) CO2 retention -- this will get worse and worse, and will likely push the patient into respiratory acidosis
(3) A right shift on the oxygen-hemoglobin dissociation curve
(4) Decreasing PaO2 which one can appreciate on a blood gas/ABG
(5) Decreasing SaO2 which one can appreciate on a pulse oximeter
