To understand this you need to remember that there was no "French Army" as there has been since the Revolution, in the sense that it was not an army of conscripts (and since the late 1990s of professional soldiers) with loyalty to the nation, but an army of professional soldiers with loyalty to whomever paid them, with a significant share of foreign soldiers, in particular in elite units. This was the normal organization for an army in the XVIIIth century, and one of the reasons why (aside procurement) war in this era was an expensive matter, involving relatively limited numbers (compared to the Napoleonic era, and modern warfare after the second half of the XIXth century) of soldiers. Think Barry Lyndon if you want to have a relatively good portrayal of the life of a soldier in a European army in the XVIIIth century. This Royal Army had different regiments which were until 1791 either Royal Regiments (Régiment du Roi, Régiment de la Reine, Royal Roussillon, etc.), Princes' Regiments (régiments d'Orléans, de Bourbon, de Condé, de Conti, that is Regiments whose colonel was a prince of royal blood), provincial regiments deriving their names from their province or the city where the soldiers were recruited (for example the *Régiment de Guyenne", for the Bordeaux area), and Régiments étrangers (Suisses, Allemands, Irlandais, etc.).
In the late 1780s, the idea of a general militia of conscripts was not something the population wanted^1, and in any case, the Army recruited its rank and file in the lowest classes of French (apart from the specific case of Régiments Etrangers, recruiting foreign professionals). Soldiers are more commonly used in law and order operations in the cities and tend to be less of a nuisance for the population as they used to be, and more integrated in the city's life^2. The officer corps remains respected, either as nobles (most were) or regarded as such by the general population. A fair share of the officer corps also took part in Freemasonry^1 ^1b, opening them to the new ideas of the Enlightenment and allowing them to share the same social mores and circles as the urban elites of the time.
The Army (apart from elite unites loyal to the king's person) had some sources of discontent : first the introduction of stricter discipline on the Prussian model (think Barry Lyndon again)^3, the restrictions imposed by Louis XVIth on promotion of non-noble NCOs to officer (restrictions imposed under pressure by the nobility)^4, and the strict social separation between a noble officer corps and rank and file.
When the first unrests in Paris began in the summer of 1789, the Army, in its non foreign regiments, largely sympathized (at least the NCOs and soldiers) with the new ideas and with the Paris population in their demands. The bulk of the rank and file of the Gardes Françaises: the largest regiment of the maison militaire du roi de France and the permanent garrison of Paris, refused to obey their officers at a crucial point in the early stages of the Revolution. Some Gardes allegedly joined with the Parisian mob on July 14, 1789 and participated in the storming of the Bastille.
The elite units dedicated to the personal protection of the King were decimated in several later episodes where they tried defending the royal family from riots, and most famously during the defence of the Palais des Tuileries by the Gardes Suisses on Aug. 10th 1792.
The final nail in the old French Army's coffin was the emigration of nobles (I mentioned in a previous post the Mémoires d'outre tombe by Châteaubriand, which discuss this period extensively), which led to the disparition during the years of 1790 and 1791 of the cadre of officers favourable to monarchic institutions, and their replacement by non-noble NCOs which were to become the core of the French revolutionary army (and later the bulk of Napoleon's Generals) : this was for example the career path for the famous Marechals Soult and Bernadotte.
Another thing you need to keep in mind is that the French Revolution was a gradual process of contestation and erosion of the Ancien Régime power structures. It was not a sudden process which went in a few weeks from absolutist royal rule to the republic, but a transition over several months with the Etats Généraux and the Constituante to a constitutionnal monarchy, where the king retained significant power and extensive prestige and support from the population, and, from that point, a slow erosion over several years of the legitimacy of the monarch, and affirmation of the assembly's authority setting in place the proclamation of the Republic in 1792 (3 years after the initial meeting of the Etats-Généraux). As such, the French Army didn't initially quell a "French Revolution" as the transformations and unrests were at first initiated by the monarchy or at least ambiguously condoned, and the objective initially shared was absolutely not an abolition of the Ancien Régime but its transformation.
^1 : https://www.persee.fr/doc/rnord_0035-2624_1993_num_75_299_4797
^1b : Le militaire en Maçonnerie (XVIIIème-XIXème siècles), Jean-Luc Quoy-Bodin, Histoire, économie & société Année 1983 2-4 pp. 549-576
^2 : Paris et l'armée au XVIIIe siècle, Étude politique et sociale, André Corvisier, Journal des Savants Année 1982 3-4 pp. 343-367
^3 : Bernard Deschard : L'armée et la Révolution. Du service du Roi au service de la Nation, 1989
^4 : La réaction aristocratique avant 1789 : l'exemple de l'armée (à suivre), David D. Bien, Annales Année 1974 29-1 pp. 23-48
^5 : La place de l'armée dans la Révolution française, André Consister, Revue du Nord Année 1993 299 pp. 7-19