Why does Bourne get headaches?

In her book Robert Ludlum: A Critical Companion, Gina MacDonald draws from The Diagnostic and Statistical Manual of the American Psychiatric Association to discuss author Ludlum’s development of the causes and symptoms of Webb/Bourne’s ailments (p 134-135).

Bourne is the deadly alter ego that Webb assumed to survive the jungle
warfare in Vietnam and Cambodia as a member of the Medusa Brigade; in
times of stress this alter ego surfaces and the gentle Webb recedes.
Unfortunately, Webb will never rid himself of the results of his
initial trauma unless he can destroy the cause of retraumatization,
specifically, the Jackal. The series explores Bourne’s progression:
amnesia and a divided personality resulting from debilitating
psychological responses to trauma, then retraumatization, treatment,
and gradual progress toward normality, followed by retraumatization,
and a final working out of conflict that may result in a reintegrated,
whole personality…

Ludlum, drawing on the expertise of relatives in the medical
profession, provides plentiful details about causes, symptoms, and
retraumatization…

Post-Traumatic Stress Disorder (PTSD) [is] a constellation of
characteristic symptoms that develop following a psychologically
traumatic event… the immediate systematic responses include heart
palpitations or tachycardia, cardiac uneasiness or pain, headaches,
giddiness, dimness of vision, and general neurological upselt
(exaggerated respiration, dilation of arteries, increased generation
of adrenalin). The long-term effects include deep anxiety, depression,
hysteria, dissociative reaction, depersonalization, neurasthenia,
intense psychological distress when exposed to events resembling the
initial trauma event,…intrusive thoughts, flashbacks, and nightmares.
There may be a feeling of detachment or estrangement from others,
difficulty sleeping, persistent irritability or anger, hypervigilance,
exaggerated startle response, guilt, episodes of terror or panic,
difficulties making decisions, and a feeling that one’s personality
has changed. These are all effects that Webb/Bourne experiences at one
time or another in the series.

So, the short answer is PTSD.