He claimed that he was on a merely religious mission, not political, but he was executed not for heresy but for treason. Questions of religion and politics were mixed because the Pope had some years earlier deposed Queen Elizabeth and forbade any Englishmen to obey her on pain of excommunication. A good article on the topic is: Papists, and the "Public Sphere" in Early Modern England: The Edmund Campion Affair in Context, Peter Lake and Michael Questier, The Journal of Modern History, Vol. 72, No. 3, (Sep., 2000), pp. 587-627 . http://www.jstor.org/stable/3079477.
The Pope's 1570 bull Regnans in Excelsis said:
Thus, anyone who was a Roman Catholic and who believed in the authority of the Pope had to, as a matter of religious belief, refuse to obey the English government. Edmund Campion, as a Jesuit, took a special vow of obedience to the Pope. Thus, insofar as he followed his religion, he was a traitor to England-- or, if you like, a traitor to Parliament and Queen Elizabeth, if loyal to Queen Mary Stuart. Further, the Bull implies that any Roman Catholic should use all efforts to obey the legitimate ruler-- Mary Stuart-- which meant to depose the pretended ruler, Elizabeth.
Therefore, resting upon the authority of Him whose pleasure it was to place us (though unequal to such a burden) upon this supreme justice- seat, we do out of the fullness of our apostolic power declare the foresaid Elizabeth to be a heretic and favourer of heretics, and her adherents in the matters aforesaid to have incurred the sentence of excommunication and to be cut off from the unity of the body of Christ.
IV. And moreover (we declare) her to be deprived of her pretended title to the aforesaid crown and of all lordship, dignity and privilege whatsoever...
V. And also (declare) the nobles, subjects and people of the said realm and all others who have in any way sworn oaths to her, to be forever absolved from such an oath and from any duty arising from lordshop. fealty and obedience; and we do, by authority of these presents , so absolve them and so deprive the same Elizabeth of her pretended title to the crown and all other the abovesaid matters. We charge and command all and singular the nobles, subjects, peoples and others afore said that they do not dare obey her orders, mandates and laws. Those who shall act to the contrary we include in the like sentence of excommunication.
In such circumstances it does not seem unreasonable to me to make it illegal for priests to enter England, or that being a priest loyal to Rome would be prima facie evidence of treason.
See also "Campion's Brag,") his challenge to debate.