Quick answer: Support from Iran via Syria and stong social presence. More detail:

• Hezbollah military men get trained in Iran. They also get weapons from Iran through Syria. Lots of weapon and artillery. They know the country and the villages and the inner streets so they are very well positioned to fight off any attack.

• Hezbollah men will fight to the death because it is a cause they are born and raised to believe in, unlike regular soldiers. Plus if they die, other than the prestige, they become martyrs and their wives and children are taken care of financially by the party.

• In addition, they created themselves as champions of the Shia sect. Shia in Lebanon were the underdog and completely neglected by the government,they had virtually no representation. So Hizbollah provided them with a voice, not to mention all the charity and rebuilding work, so Hizbollah had a major popular base.

• Hizbollah's leader, Nasrallah, is a seriously intelligent and charismatic man, he has a large appeal in the country even among some of the Christians.

• The first thing is that Hezbollah emerged essentially in fertile ground. Lebanon had been experiencing civil conflict for a while before Hezbollah appeared on the scene, so they weren't just some random guys getting together, they could draw on a degree of experience.

• Reinforcing this, Hezbollah received training in Iran and from cadres from the Islamic Republican Guard Corps, which is to this day well regarded for its skill for irregular warfare etc. As a side note, a question I wouldn't mind being answered is just how the IRGC got to be so good at its job. AFAIK it didn't really have an Iranian pre-cursor, and while it might have picked up some knowledge from what remained of SAVAK etc. this would seem strange at the IRGC was set up essentially as a counter to the 'old regime' (specifically the army), which would make their precursor being the Shah's secret police seem odd. Anyhow, I digress.

• Part of it was also an evolutionary pressure thing - Hezbollah had to be good to survive. It emerged in an area under Israeli occupation, and to add to the pressure it found itself in conflict with other Lebanese paramilitary groups like Amal (another Shia group) in the War of the Camps. It might be notable that Hezbollah sided with the PLO in that conflict - perhaps Hezbollah received assistance (training, funding etc.) early in the piece from Palestinian militant groups?

• In terms of finance, Hezbollah was receiving support from Iran, Syria, Lebanon's large Shia population and also the Lebanese diaspora. And no, it isn't just drug money. Hezbollah's economic portfolio is pretty diverse and not all of it is illegal either, they have their own construction companies for example.

As to how it got to be stronger than the Lebanese army, part of that isn't why Hezbollah is strong, and more about why the Lebanese army was weak. The army had been deliberately kept weak by Lebanese governments for fear it would be used by one ethnic group or another to seize power or whatever. Then with the onset of civil war and successive invasions, occupations etc. the army was hurt badly. Even if the government had had a change of heart and wanted to build up the military, this would have been difficult, since the economic base of Lebanon had been badly hurt by the civil war and occupation. So post occupation, you find an army still trying to find its feet while Hezbollah is now 'free', a leaner and meaner force that wasn't necessarily short of heavy weaponry due to its foreign backers.

*Editted for typos... I hope I got them all this time