Fernando Maximiliano José María de Habsburgo-Lorena, otherwise known as Maximiliano de Habsburgo for us mexicans, was, it should be noted, never the brightest bulb in the Habsburg shed, the exact reason he accepted his participation in the Second French Intervention are never explicity said (beyond banalities about his oh so infinite love for a country he didnt knew or had ever set foot on before the invasion), but there are a few details about his life previous to the invation and details in the Treaty of Miramar (by which all involved parties agreed to the invasion) that shed some probable reasons.

First of all, Maximilian himself wasnt a rich or particularly important in Austrian politics, oh, he wasnt poor, he had a healthy salary as Commander in Chief of the War Fleet -a position he occupied at 22 years old having no battle or highseas experience-, but his main source of money was really from his marriage to princess Carlota Amalia of Belgium (Belgium would see mayor involvement in the subsequent invasion including dispatching troops and negotiating to obtain mayor minning consecions in the country) in 1857, but he himself held little productive land or important titles, and -importantly-, he wouldnt inherit any either, with his brother Franz Joseph I being the emperor of Austria and the direct inheritor of the imperial line.

Of course this doesnt mean that he never attempted to obtain political charges that would allow him to stand on his own, which leads us to the 2nd point, thanks to the resources from his marriage to Carlota he was able to... errr, obtain the governorship of Lombardy-Venetia, which he then endeavored to adminster, within a year, the entire province was on open revolt and about to achieve independence from the Austrian Empire, he was quickly replaced by strongman Franz Gyulai, and whether justly or unjustly he was quietly retired from any positon of authority.

So, in 1861 Maximilian of Habsburg is a somewhat impoverished (for an austrian royal) noble who has no hope of ever ruling over anything more that a garden, this is were José María Gutiérrez Estrada enters the scene, I will touch on Napoleon III and his wife Eugenia de Montijo in a bit, but for a moment I want to draw your attention to 2 points that Gutierrez agreed with Maximilian on the so called Convention of Miramar that I assume any impoverished Autrian noble would've also taken notice of:

4 - A loan of 100 million dollars (to be paid by mexicans). For the payment of interests of this debt of 5%, can be offered in mortage of the clerical lands that havent yet been sold off and which value can be calculated in between 20 to 25 million dollars...

To give you an idea of what this ammount represented, when president Benito Juarez exited Mexico city at the height of the invasion, the national budget was around 400,000 pesos. And secondly:

7 - To guarantee the service of conservative chiefs and other parties (they had called on such characters as Gral. Santa Ana) it will be of the outmost importance to have on hand an ammount no less than 200,000 dollars (secured by France) the ammount of which will be able to be forwarded on the enterprise.

Added to that were the tall tales of the mexican conservatives, of Mexico being the horn of abundance were unlimited wealth (which Maximilian would have at his disposal) would be found with modern cities, infrastructure and ammenities everywere like a european country (this part would be the biggest shock to Maximilian according to Juan Blasio, his personal secretary, in fact one of the first mayor expenses of Maximilian in Mexico would be to comission the construction of a theater so he could enjoy some opera).

The French

Now, on the side of the French there were their own interests, remember Maximilian wasnt the conservative's first choice (even the house of Moctezuma was approached at one point but they were turned down), Napoleon III had plans though, and this fitted perfectly on his political agenda, its a little bit long but I think this correctly summarizes French interest in the region, to quote historian Doralicia Carmona Dávila (translation by me):

... invading Mexico, a country which, according to french senator Michel Chevalier had huge gold deposits in Sonora. Was also attractive for the construction of a channel in the Istmus of Tehuantepec that would join both oceans. Napoleon III wished to create a Latin League that would include mediterranean land and former portuguese and spanish colonies. A french bastion in Mexico, could be the basis to impose protectorates in central and south America. Years before, in 1854, french count Gastón de Raousset-Boulbon had paid with his life his attempt to create a new country in Sonora, Sinaloa, Durango and Chihuahua. Now it was the right time with the economic chaos in Mexico and a US distracted by the Secession War. To establish a government tied to France, would allow Napoleon III to build a french, monarquist, catholic force in the region to upset the influence of saxon, republican, and protestant influence of North America and also obtain debt by the mexicans whose main debt holder was his own brother the Duc of Morny. It was Eugenia de Montijo, the emperor's wife who suggested Maximilian... this would also serve as an olive branch to the Austrian emperor after the war of 1859 which led to the Italian reunification.

So, TL;DR its possible that Maximilian of Habsburg very likely saw an oportunity to obtain riches and power the likes of which he would never obtain in his home country and was the only person err.... hopeful -let us say it like that- enough to undertake such a gamble.

I would also encourage you to consider reading Maximiliano Intimo, it was written by his personal secretary Juan Blasio, he attempts desperately to defend Maximilian in the book, but... yeah, as a mexican myself its a fun game to read the book and take a shot everytime foreign interests, spending lavishly on dances while the troops need resources to fend off Juarist counter-attacks, executing revolting mexican civilians or pilling up debt on Mexico itself are mentioned in the book.

Wow. Excellent question and answer. Now I have to research this. I have never heard of this whole scenario!