Boswellia serrata Roxb. extract (BSE), rich in boswellic acids, is well known as a potent anti-inflammatory natural drug. However, due to its limited aqueous solubility, BSE inclusion into an appropriate carrier, capable of improving its release in the biological target, would be highly desirable. Starting with this requirement, new hybrid composites based on the inclusion of BSE in a lamellar solid layered double hydroxide (LDH), i.e., magnesium aluminum carbonate, were developed and characterized in the present work. The adopted LDH exhibited a layered crystal structure, comprising positively charged hydroxide layers and interlayers composed of carbonate anions and water molecules; thus, it was expected to embed negatively charged boswellic acids. In the present case, a calcination process was also adopted on the LDH to increase organic acid loading, based on the replacement of the original inorganic anions. An accurate investigation was carried out by TGA, PXRD, FT-IR/ATR, XPS, SEM, and LC-MS to ascertain the nature, interaction, and quantification of the active molecules of the vegetal extract loaded in the developed hybrid materials. As a result, the significant disruption of the original layered structure was observed in the LDH subjected to calcination (LDHc), and this material was able to include a higher amount of organic acids when its composite with BSE was prepared. However, in vitro tests on the composites’ bioactivity, expressed in terms of antimicrobial and anti-inflammatory activity, evidenced LDH–BSE as a better material compared to BSE and to LDHc–BSE, thus suggesting that, although the embedded organic acid amount was lower, they could be more available since they were not firmly bound to the clay. The composite was able to significantly decrease the number of viable pathogens such as Escherichia coli and Staphylococcus aureus, as well as the internalization of toxic active species into human cells imposing oxidative stress, in comparison to the BSE.