Inulinases are enzymes involved in the hydrolysis of inulin, which can be used in the food industry to produce high-fructose syrups and fructo-oligosaccharides. For this purpose, different Aspergillus strains and substrates were tested for inulinase production by solid-state fermentation, among which Aspergillus terreus URM4658 grown on wheat bran showed the highest activity (15.08 U mL−1). The inulinase produced by this strain exhibited optimum activity at 60 °C and pH 4.0. A detailed kinetic/thermodynamic study was performed on the inulin hydrolysis reaction and enzyme thermal inactivation. Inulinase was shown to have a high affinity for substrate evidenced by very-low Michaelis constant values (0.78–2.02 mM), which together with a low activation energy (19.59 kJ mol−1), indicates good enzyme catalytic potential. Moreover, its long half-life (t1/2 = 519.86 min) and very high D-value (1726.94 min) at 60 °C suggested great thermostability, which was confirmed by the thermodynamic parameters of its thermal denaturation, namely the activation energy of thermal denaturation (E*d = 182.18 kJ mol−1) and Gibbs free energy (106.18 ≤ ΔG*d ≤ 111.56 kJ mol−1). These results indicate that A. terreus URM4658 inulinase is a promising and efficient biocatalyst, which could be fruitfully exploited in long-term industrial applications.