Exegetical Commentary on the  Code of Canon Law (PDF pp. 3271-2) describes what obsequium* religiosum means:
*p. 916 (PDF p. 948) of New Commentary on the  Code of Canon Law: "An exact translation of obsequium is difficult, but 'submission' is not the best one because it exaggerates the force of the Latin. Such English terms as 'respect,' 'deference,' 'concurrence,' 'adherence,' 'compliance,' or 'allegiance' would be better translations of obsequium."
2. Clarification of the response of the faithful using positive and negative terms
If classifications of the magisterial acts being studied here cannot avoid using negative terms to distinguish them from infallible magisterium, the same thing arises in describing the attitude with which the intellect and will of the faithful are to receive teaching presented in a universal and non-infallible magisterium: the faithful are to accept it with religiosum obsequium, which is not exactly the same as the assent to the divine and Catholic faith with which the faithful, under c. 750, are to accept the infallible acts of magisterium described in c. 749.
In order to grasp the exact meaning of obsequium religiosum, which is the [new] term used by Vatican II (LG 25) to describe how the faithful should receive teaching expressed in the acts of magisterium in question, it is necessary to appreciate both the similarity and differences between obsequium religiosum and obsequium fidei.
The Church, since St. Paul's teaching in 2 Cor 10:5-6, which describes the faithful's personal response to God who reveals Himself as "in captivitatem redigentes omnem intellectum in obsequium Christi ['bringing into captivity every understanding unto the obedience of Christ']," has found no better way to express the loving and submissive acceptance by the intellect to divine revelation and to the dogmas of the magisterium than to repeat the Apostle's words, i.e., by plenum intellectum et voluntatis obsequium. By this reception, the obedience and submission of the intellect and will to revelation, "se totum libere Deo committit ['freely commends all to God']" (DV 5), and the body of the faithful glows brightly "per commune omnium obsequium erga Ecclesiæ mysterium ['through the common obsequium of all with respect to the mystery of the Church']."
Unlike obsequium fidei, obsequium religiosum, with which the faithful are to receive doctrine expounded in non-infallible authentic magisterium of universal scope, means that it "must be accepted reverently as supreme magisterium and the faithful must sincerely obey it in their intellect and will" (LG 25).
As can be seen, sincerely obeying with one's intellect and will does not imply "bringing every thought into captivity to the obedience of Christ" (2 Cor 10:5-6) as does obsequium fidei. Yet acceptance of this authentic, universal, and supreme magisterium as such requires obedient assent, i.e., a loving and submissive acceptance by one's intellect, will, and conduct, "with one's mind and will in accord" (LG 25) with a non-infallible act of magisterium. Obsequium religiosum "cannot be just external or a matter of discipline, but must be seen with logic and from the perspective of faith."8 Doctrine expounded in these magisterial acts is not only consistent at all times with doctrine taught in dogma, but also, by virtue of divine assistance with which these magisterial acts always have an intimate relationship with the truth. Therefore, the hallmark of religious assent is obedience.
Religious assent recognizes that the universal non-infallible magisterium exercises a divinely assisted function, that it protects the deposit of revelation by promoting and unifying its teaching and application to certain historical and cultural circumstances. The constant flux in circumstances ensures that the type of magisterium described here (unlike statements of dogma) cannot and should not avoid so many historical, local, and ephemeral references. This is the "original limitation"10 of the non-infallible magisterium and at the same time its special usefulness, so that each Christian generation may find up-to-date guidelines to direct their apostolic acts in their own times and places.
Obsequium religiosum does not mean one is free to dissent:
In contrast to the obedient assent to be given this type of magisterium in submission and love, there have been a number of opinions over the last several decades that take the "original limitation" of this type of magisterium as grounds for an alleged right of the faithful to dissent from such a magisterium. By equating "original limitation" with an alleged possibility of "error," they claim that the faithful may dissent from it out of an alleged maturity acquired by better and surer ways. The model of protesting governmental authority taken from civil society would express the attitude of the faithful to this type of magisterium.
This perversion of obsequium religiosum is not acceptable even for an attitude on the part of those who dedicate themselves to the sacred sciences whose just liberty to do research and express an expert opinion in their areas of expertise should still be duly submissive to the magisterium of the Church (c. 218). This type of magisterium is still open to possible definitive statements in the future and needs the special contributions of students of the religious sciences whose assessments, made in accordance with the principles and methods of each discipline, are awaited and invited by the magisterium itself, "received with an accepting mind and will" (LG 25; cf. VSp 109-113).
cf. Humani Generis §20, quoted in this answer to "What does 'authentic magisterium' entail in Lumen Gentium?"