Ok, so before I get around to your main question (I promise I will!), we need to take a step back for a bit and look at your underlying assumption - "the [early Capetian] Kings of France supposedly had little power". This is, admittedly, the traditional view: as France descended into 'feudal anarchy', it was each (war)lord for himself, and the Capetian kings found themselves surrounded by powerful counts and dukes who were independent in all but name. These 'over-mighty vassals' pretty much ignored the king and certainly limited his influence, military or otherwise, outside of a tiny island of lands, the royal demense, around Paris and Orléans.

In the last ten or fifteen years however, historians have really started to serious question the validity of this interpretation. The reasons why become quite clear when you examine the reign of the penultimate Carolingian king of France, Lothar (954-986). Pretty much the entirety of Lothar's reign was a struggle to contain the influence of his cousins, Hugh, Otto and Henry, the sons of Duke Hugh the Great (d. 956). Although it had its ups and downs, the end result was that the eldest son, Hugh Capet, won decisively over the king; by c. 980 Hugh Capet's lands and influence stretched so far and wide that the influential cleric (and future pope) Gerbert of Aurillac wrote that it was Hugh who was king in all but name in Lothar's kingdom. When Lothar died in 986 and his childless son Louis V died just a few months later, Hugh made the transition from duke to king and founded a dynasty that would rule France for the next 800 years.

So how did the most powerful family in France go from their position of strength to their characteristic weakness by gaining the throne? The short answer is, they didn't. It just looks that way.

The early Capetians (usually considered to include Hugh Capet, Robert II, Henry I, Phillip I and Louis VI) weren't necessarily weaker than previous kings were. It's hard to point out any spectacular failures or weaknesses on their part - even if Henry I did have his ass handed to him by William the future-Conqueror in the 1050s. When they wanted to, the Capetian kings could and did enforce their authority through military means. This is perhaps most spectacularly seen in Robert II's 'conquest' of Burgundy, which was actually 2 separate (but linked) campaigns a decade apart to quell political opposition in the duchy after the death of his uncle Duke Henry in 1003. But what sees so un-kingly and characteristic of weakness to us is that Robert more or less gave Burgundy away immediately, ceding autonomy and large stretches of territory to people like Landric of Nevers and Otto-William of Macon, who had rebelled against him only a few years earlier. He gave the duchy itself (i.e. the title of Duke and the ducal lands) to his second son Henry, in order to found a separate branch of Capetians (after the death of his older brother Hugh, Henry became next in line to the throne and passed the duchy to his younger brother Robert, who founded the Capetian line of Burgundian dukes). But direct military intervention like this was rare and the fact that the Capetian kings rarely did it, despite the constant squabbling, rebellions and infighting of the great lords of France.

The real explanation for Capetian 'weakness' then, isn't actually a weak monarchy, but a monarchy that chose not to wield power in the same way as their predecessors. As former aristocrats themselves, the Capetians saw themselves as primi inter pares, firsts among equals, and apparently saw little need to intervene directly in the affairs of the other great lords, unless there was a real need. The Capetians worked hard to build consensus and networks of alliances to control France, rather than enforcing rigid central control - this was a tremendous balancing act, involving a constant stream of negotiation and compromise. The sorts of political communication involved can be seen in the correspondence of Bishop Fulbert of Chartres, who often worked as an intermediary and arbitrator between Robert II and his most powerful lords. Here we can see the intangibles of royal power: for example, in a dispute between the counts of Anjou and the dukes of Normandy, both plead with Robert to protest the righteousness of their cause, to sway him one way or the other. They're not asking him to make a ruling or to micromanage, but they recognise his authority and his role, by picking sides, to keep the balance of power in France. And, by and large, the early Capetians did this well. When the decades long alliance between the monarchy and Normandy fell apart in the 1050s, the Capetians found support in Flanders and Anjou to keep William the soon-to-be-conqueror in line. When Philip I faced papal and episcopal censure for his alleged bigamy, he had a strong enough system of allies amongst the nobliity (including from Fulk IV of Anjou, whose wife he stole) and, crucially, from the bishops.

The religious element of kingship was probably the biggest element which distinguished the royalty from the other great lords. The king was chosen by God and consecrated with holy oils at coronation, just like a priest. Ecclesiastical support was vital for political success (we already saw Fulbert of Chartres' role as a political intermediary). The Capetians carefully cultivated their religious image and patronage: royal monasteries like Fleury and St Denis were given valuable lands and privileges, and in return wrote propagandist works in honour of the kings (Helgaud of Fleury's Life of Robert the Pious and Suger of St-Denis' Deeds of Louis the Fat). Kings had a strong influence over episcopal appointments too and often worked to get close allies or even family members into prominent ecclesiastical office: Hugh Capet's half-brother Heribert was bishop of Auxerre, and his illegitimate son Gauzlin became Archbishop of Bourges. Schemes for spiritual legitimacy could go badly wrong however, usually due to royal familiy members' personal failures: we already saw Philip I's bigamy, and his brother Hugh of Vermandois' infamous desertion of the First Crusade in 1097 damaged the family image so much that Philip married his daughter to the crusader hero Bohemund of Taranto, Prince of Antioch, in 1106.

So, in summary, what made the early Capetian kings of France 'weak' was actually a choice not to be like other kings, and more to act more like their great lords. But they were different, both in what was expected of them and in their actions: they were seen as the biggest and strongest lords, with a responsibility to ensure order and balance, backed up with the spiritual authority of the monarchy.