Did Peter the Great promote rights for women? If so, how?

The short answer is he didn't, not really. By "participated more actively", your source likely just means socialisation, rather than women's rights or activity in general society. To be sure, some of his policies had a positive impact on the welfare of women, but that is more incidental than intentional.

Basically, it is a bit of a stretch to describe his reign as "took care of" women's rights.

Social Interactions

As is well known, Peter the Great implemented sweeping modernisation reforms in Russia, inspired by the Enlightenment. He was heavily influenced by the West in this endeavour, and accordingly sought to introduce much western cultural and social norms.

One part of the Petrine social engineering was the introduction of new fashion styles. Both men and women were ordered to adopt western wardrobes. Specifically for women, in 1700:

Peter ordered the women as well to adopt Western dress, giving them until January 1, 1701. The aristocratic women in particular disliked the new dress, with garters to hold up the stockings, high-heeled shoes, low necklines, and worst of all, uncovered hair.

_{- Bushkovitch, Paul. Peter the Great: The Struggle for Power, 1671–1725. Cambridge University Press, 2001.}

More substantially, this set the stage for Peter to create a Western-style court. He accomplished this by bring the traditionally secluded Russian noblewomen into the social life at his capital.

Peter breached the walls of the terem, forcing elite women to leave its seclusion in order to socialize in public at European-style evening parties ... they were expected to perform Western dances, to display appropriate social skills, and to converse with men in French.

_{- Engel, Barbara Alpern. Women in Russia, 1700-2000. Cambridge University Press, 2004.}

Peter promoted such socialisation beyond the royal court, as well. In 1718 he issued an edict on "assemblies", public social functions, which were to invite women.

Peter hoped to introduce refinement and rational ways fo passing the time to his countrymen. The 1718 decree cited French practice, and orderd the nobility to put on something between an open house and a ball ... The "assembly" had to include women, a break with the traditional all-male banquets and drinking parties.

_{- Bushkovitch, Paul. Peter the Great: The Struggle for Power, 1671–1725. Cambridge University Press, 2001.}

Marriage

Another aspect of the Petrine reforms concerned marriage. Peter broke the customary monopoly of Muscovite parents in arranging marriages for their children. Whereas before, brides and grooms might only meet for the first time on their wedding night, now Peter demanded freely given consent from both parties.

[I]n 1724 the Synod promulgated the tsar's decree requiring that before the marriage of their children or servants the respective parents or guardians or masters must swear and oath that they consented to the marriage, that the persons about to marry 'ardently' wished to do so, and that no form of pressure had brought to bear on them.

_{- Cracraft, James. The Church Reform of Peter the Great. Stanford University Press, 1971.}

Additionally, Peter forbade women from taking the veil before the age of 50. This prevented fertile women from wasting their reproductive potential; but it also meant that husbands could no longer dispose of inconvenient wives by forcing them to enter monasteries (as Peter himself had done).

Conclusion: Not that Rosy

Some of these reforms have inspired writers to speak of an emancipation of women under Peter the Great. In reality, care should be taken to not overstate this.

The Petrine Revolution ended the seclusion of noble women and benefited some others in very specific ways. It did not particularly promote women's rights or welfare in general. The subordinate status of women remained within contemporary European molds.

Authority remained in the hands of husbands and fathers ... women enjoyed no independent civil status. As was the case elsewhere in Europe, a woman's status was determined by the rank of her husband or father. The law continued to deal far more harshly with women than with men.

_{- Engel, Barbara Alpern. Women in Russia, 1700-2000. Cambridge University Press, 2004.}

The 18th century did see a gradual recovery of women's property rights in Russia, but few of this could be unequivocally credited to Peter's reign. For example, Peter's 1714 Law of Single Inheritance actually prohibited women from receiving lands as dowries. Moreover, while women were theoretically empowered to control their own dowries, the subordinate status of women in civil life meant that their husbands could exercise de facto control.

[H]usbands [in practice] frequently could force their wives to comply with their wishes. Court cases from the eighteenth century recount many instances in which husbands beat or tormented their wives to force them to agree to mortgage to sell their dowries or dissipated their wives' dowries without their knowledge.

_{- Pushkareva, Natalia. Women in Russian History: From the Tenth to the Twentieth Century ME Sharpe, 1997.}

Similarly, Peter the Great did not promote the employment of women outside traditional roles. A woman's role in Petrine Russian society remained domestic, with her main responsibilities to the family. The reproductive emphasis did cause women's education to be deemed important for the betterment of the next generation. However, proposed schemes to educate women all amounted to nothing under Peter's reign.

Peter even proposed to send noblewomen abroad to learn languages and social graces. But in this case, fierce parental opposition forced him to retreat. An edict of 1722 required basic literacy of women by forbidding a woman to marry if she could not sign her name. No one appears to have paid attention.

_{- Engel, Barbara Alpern. Women in Russia, 1700-2000. Cambridge University Press, 2004.}