Let's set the stage for the significance of this line, which can only be understood knowing what has occurred before and what occurs after. Giovanni is a young student who takes lodgings in rooms that were the top floor of what appears to be a gloomy old mansion. An old woman, Lisabetta, who may be a servant, a concierge, or Giovanni's landlady (this is not clear), points out to Giovanni the view from his window, which is a magnificent and flourishing garden, the garden of Doctor Rappacini, who, Lisabetta tells Giovanni, is said to create potent medicines from what he grows. She also tells Giovanni that Rappacini has a daughter, whom he might see from time to time in the garden.
Giovanni continues to look at the garden as time goes on and one day observes an elderly man tending the garden. He notices that the man avoids any direct with anything growing in the garden,
...with a caution that impressed Giovanni most disagreeably; for the man's demeanor was that of one walking among malignant influences, such as savage beasts, or deadly snakes, or evil spirits, which, should he allow them one moment of license, would wreak upon him some terrible fatality (para. 9).
Giovanni finds this odd and even frightening as he thinks to himself that this garden is a perversion of the original Eden and the old man a very odd sort of Adam.
As he is contemplating this, he sees the old man approach a plant and place a mask on his face, with which he is apparently not satisfied. The old man calls "Beatrice," who appears in the garden, a beautiful young woman who is the old man's daughter. She is able to approach this plant and even to embrace it, clearly without risk of harm.
Giovanni dreams of the girl and the garden, and the next day, when he visits a professor of medicine from his university, Professor Baglioni, he learns that Doctor Rappacini has the reputation of caring far more for the pursuit of knowledge than he cares for anything else, willing to sacrifice human life in the name of science. Giovanni also learns that having been trained by Rappacini, Beatrice is as brilliant as she is beautiful, and all the young men are in pursuit of her.
Intrigued and infatuated already, Giovanni buys a bouquet of flowers for Beatrice, and he lingers at his window, hoping to see her. He observes Beatrice moving freely amongst the flowers, finally plucking one and pressing it to her bosom, not only with no ill effects, but also seeming to bring her energy and health.
This is when Giovanni sees the drops fall upon the lizard. And then he sees that,
For an instant, the reptile contorted itself violently, and then lay motionless in the sunshine (para. 32).
A few minutes later, he observes an insect flying into the garden, also to die, at Beatrice's feet. And his bouquet, which he tosses to her, starts to wither as she goes back into the house.
These are all warnings to Giovanni, Rappacini's avoidance of the plants, the cautions of Professor Baglioni, the death of the lizard, the death of the insect, and finally, the death of his bouquet. And in the midst of all that toxicity and death, Beatrice is clearly thriving.
All of these warnings are a form of foreshadowing for the reader, who understands, quite likely before Giovanni does, that Professor Rappacini has done something to his daughter to allow her to be in this garden, with the suggestion that not only can she coexist with these plants, but also that they somehow nurture her.
As the story unfolds, we see this is what has happened. Giovanni becomes embroiled in Dr. Rappacini's machinations as he falls more and more deeply in love with Beatrice. She attempts to protect him from the poisonous garden, and Professor Baglioni tries to warn him of the danger of his love, for because of Rappicini's "experiments," Beatrice herself is poisonous. Giovanni, with the help of a potion from Professor Baglioni, plans to "cure" Beatrice and rescue her from her father. He sees that he himself has become toxic, killing off ordinary flowers in his own presence, and he grows more determined now to rescue both of them. As Beatrice takes the potion, Rappacini appears, appearing to lament the loss of his work, rather than the loss of his daughter. Beatrice dies at the feet of her father and Giovanni, from the potion, which releases her from the evil her father has wrought, and we are left to imagine Giovanni going on, a sadder but wiser young man.