On a general scale, Christianity was successful in replacing or absorbing other previous cults and religions for a number of reasons, if we exclude the specific cases in various contexts.
Here I will attempt to summarize what I've come to understand.

Its strongest feature, as far as I have been able to comprehend, was its structured nature and existence of an established canon of scriptures and teachings, which for the time most proselytizing happened within Europe (VI-X centuries) had already been worked on for 400 years by theologians and scholars such as Saint Augustine, Saint Ambrose, Saint Jerome and pope Gregory I, and the other who elaborated what would be come to be accepted as canonicity (to the point where we usually distinguish between the Church Fathers before, antenikeans, and after, postnikeans, the first complete effort of doctrinal consolidation with the Council of Nikea (325 AD).
Preaching the concepts and virtues of Christendom was way easier, in my opinion, when being provided by extensive texts of thoughts and suggestions over practical, real matters, if compared to the most mystery religions and cults which most likely lacked such structural and doctrinal stability. To make a quick example, st. Augustine was questioned by two early Christian soldiers if it was correct to kill as their duty to the Roman state, or they should desert and flee because it contrasted with their beliefs. Similarly, in his treatise De Genesi ad litteram ("On the literal interpretation of the Genesis), he stated that the Book of Genesis' description of the size and shape of the world were to be taken as a metaphor, not literal truth, because the reality had already been discovered by the Classical scholars and philosphers such as Hecathaeus of Myletos (the Greek who calculated the circumference of the Earth in the V century B.C., missing just a few hundred kilometers).
I seriously doubt that a druid in VI century Britannia could argue over the virtues and principles of his gods as easily as a priest or preacher could explain the cardinal virtues synthesized in the Parables of Jesus.

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Virtues and principles which could be adapted to contain previous aspects connected with other deities, as we have seen with the transition between the Graeco-Roman polytheism to early Latin Christianity with the correspondence between Jesus' birthday and the Sol Invictus festivals related to the Winter Solstice. Someone has argued that the cult of saints and the Holy Trinity within the Church, element which has caused a severe rebuke by the other two Abrahamitic religions of Judaism and Islam, condemned ironically as polytheism, was an evolution of heroic cults present in the eastern regions of the Empire (the place of origin of Christianity), vestigies of the previous Greek tradition which had bled within the Roman culture as well. This, I hypothesize, could have helped in the conversion of the first Germanic (or barbarian) populations such as the Franks and the Langobards. I am inclined to think this due to their very strong faith towards st. Martin, in the case of the Franks and st. Michael in the case of the Langobards.
Saint Martin of Tours was seemingly a member of the Schola Palatina, the élite heavy cavalry bodyguards of the late Roman emperors and served in military campaigns during the IV century, possibly as a member of a squadron of cavalry of the Equites Cataphractarii Ambianenses, according to the Notitia Dignitatum which tells it was stationed in nowdays Amiens.
Archangel Michael was the leader of the armies of God which clashed with Satan and the Fallen Angels during the War in Heaven described in the Book of Revelation. In Christianity, he has four roles, the aforementioned leader of the Heavenly Host, other two are at times associated with death, as he is both the carrier of souls and the one who weighs the deceased's soul, explaining his depiction wielding scales. Lastly, he is the defender of the Church, the sword of the Church which slays evil and darkness (one of the most common depictions of said archangel is him slaying a serpent or dragon, usually symbolizing the Devil). I still remember a prayer from mass which asks for the protection of Saint Michael.
These two "warrior saints" were very much relatable to the warlike conferedations of tribes settled or settling the imperial territories and more easily picked up instead of ancestral deities. But again, this is my personal theory so you are free to completely ignore it.

Additionally, Christianity has been a communitary religion from its inception, one that included the whole believers in its offices and practices which were part of its structure. Most religions that we know of were mysteric in nature, meaning that the aspects of faith were almost entirely known and kept by its priests alone and only those with the initiation could attain salvation. The contact with the divine was not extended to the faithfuls but mediated by an initiated figure. Responses from oracles and mystics were cryptical and confusing.
Every Christian could pray to God, ask for guidance and protection and trust in his will for the afterlife, petition him for answers and he was fully capable of receiving them without secret knowledge. All memebers of the Church who had taken part to the rites of baptism and upheld a rightous life could attain salvation. This one fact has created the hypothesis that religions centered on warrior afterlife, like the Norse religion, saw most of the non-warrior members of society joining Christianity due to this "guarantee".

Last but not least, I am sure that the care and support ecclesiastical structures provided for the communities they lived in played a key role. Churches and monasteries held wealth and were compelled to help in times of need, which are aware was an obligation very often pursued, from delaying payments of rents, to providing food in times of meagre harvest to sheltering and providing for the travelers, the sicks and the poor, a prerogative that monasteries were bound by law to abide to. Ecclesiastical lords and ladies (bishops, abbots and abbesses) also provided protection, as they could establish nets of vassalage with lay aristocrats which would be called upon for war or, as it would be the case during the end of the X century, to protect roads, churches, markets and other places of common interest and to defend the pauperes, meaning the poor but also the women, the children and the clergy.

It was a long answer to a very big question. I hope to have provided a point of view.