How can some residues in the active site of enzymes be protonated with a pKa < 7?
Generally speaking, even though physiological pH is generally ~7.4, this is not necessarily the pH inside the enzyme active site. The enzyme's catalytic machinery may experience a completely different pH for one of two reasons: 1) because the enzyme is located in a place where the pH is simply not 7.4, and 2) The enzyme's amino acid composition shifts either the pH in the active site or the pKa of the active site residues relative to the surrounding environment such that the effective local pH in the catalytic site is lower or higher than 7.4. 

For example, there are places in humans, such as the stomach or around muscle cells during anaerobic respiration where the pH is actually acidic relative to the standard physiological pH of 7.4, due to the presence of HCl or lactic acid respectively. Additionally, certain subcellular organelles, such as lysosomes, also are relatively acidic, since they are a site for breaking down proteins into amino acids. 

However, the case presented in this paper seems to be an example of the second situation, where an enzyme that is in a pH neutral environment relies on one or more protonated amino acids to perform catalysis. This is relatively common. Basically, the enzyme's active site has an amino acid composition that when folded correctly, alters the local pH around the catalytic residues to favor catalysis. For example, in the enzyme acetylcholinesterase, the active site serine is rendered nucleophilic by being positioned next to a glutamate and a histidine, in an arrangement referred to as a catalytic triad. The glutamate is an acidic residue, and its close proximity to the histidine means that it will stabilize the protonated form of the histidine. Energetically, this makes protonation of the histidine very favorable. This causes the histidine to take a proton from the active site serine. Now that serine's oxygen is deprotonated, it can act as a strong nucleophile. After the serine forms a covalent interaction with the substrate, the histidine will facilitate a similar nucleophilic attack by a water molecule, regenerating the enzyme and completing the catalytic cycle. 

Something similar is seen with the beta-lactamase TEM-1. In this case, TEM-1 uses a lysine (pKa ~11) instead of histidine (pKa ~6), so multiple acidic residues are in the active site to facilitate the deprotonation of the lysine during the catalytic cycle. In effect, the amino acid composition of the active site allows for the shuffling of protons needed for catalysis. 

So to summarize, while physiological pH is widely regarded to be 7.4, an enzyme's catalytic machinery may experience a very different pH through either the cellular/physiological localization, or through the amino acid composition of the active site. In the latter case, the presence of acidic or basic residues can alter the protonation state of other acidic or basic residues, meaning the effective pKa of these residues or the effective pH around them is higher or lower than 7.4.