What explains the existence of a lag, and why has the time delay between the recovery of output and the recovery of employment been increasing in recent recessions?
There are (at least) three factors that come into play. First, when firms see the initial signs of an upturn they ask themselves whether the change is permanent or transitory. If the upturn in the data is a false signal, a one or two month temporary upward movement in the data only to be followed by a return to recession conditions, firms do not want to make a commitment to hiring new workers. They aren't, in general, fully using the labor they already have, so they wait until they are fairly certain it's a true recovery and not a false start before hiring. Thus, there is a delay between the point in time when output turns upward and the time period when firms begin hiring new workers.
Which brings us to the second reason. Firms do not want to let their highest productivity workers, or workers that require a substantial investment in training costs, go in a recession even if there's not enough work for them to do. These workers will be needed when things turn around, you may want to retain a star in the sales department even if recent sales are relatively low. In addition, laying these workers off risks losing them permanently, perhaps to a rival firm, and having your best workers end up being employed by a rival is an outcome firms would rather avoid. In addition, when there are substantial training costs, it may be more costly to let workers go and then, once things turn around, to hire someone new that requires retraining. When firms retain such workers -- we call this labor hoarding -- they will not need to hire new workers until the workers they have are working to capacity, and given that these workers don't have enough to do during the recession, the rehiring won't begin until well into the recovery.
Third, during downturns it's natural to reorganize production, particularly when some workers don't have enough to do. Firms may lay some workers off and reassign work to those who are still there, and they will install labor saving equipment in an attempt to cut costs which may require them to let long-time workers go, something they avoid in better times but are more motivated to do when conditions deteriorate. After reorganizing, they may find that, with a new computer or piece of software, or some piece of equipment, or through the reorganization itself, one worker can do quite a bit more than they realized. Thus, having discovered how to reorganize to increase productivity, the demand for labor on the upside will be smaller than the amount lost on the downside thereby causing sluggishness in the recovery of labor.
Finally, why has the lag between the upturn in output and the upturn in employment extended in recent years? One reason may come from increased labor hoarding. As we have lost manufacturing jobs and turned to producing other types of goods and services, and as computers and other technology has entered the workplace, the specialized training and expertise it takes to do many classes of jobs has increased substantially. Because of this, the amount of labor hoarding has gone up (i.e. because the cost of letting workers go is higher, firms retain more workers who are not fully utilized than before) and that has extended the time until firms need new workers once the economy has recovered.